So, this is lecture 9 right, okay so we have been looking at polynomials over a general field right, fx, okay so this is the set of polynomials, polynomials with coefficients from a field f, okay so we have been looking at these polynomials and the last thing I did was show you, show you an example of division, how do you divide a polynomial a of x by b of x to get a quotient and a reminder and we saw how, I think I gave an example in f5 right, just to show you that it carries over from your familiar real and rational fields as well, if you have a finite field also, you could think of polynomials over that finite field, okay, alright so let's keep moving ahead with that, so the division, how did the division work, let me once again remind you real quick, the division worked as follows, so if you were to divide a of x by b of x, you would get a quotient q of x and a reminder r of x, all of these case will lie in fx as well and the degree of r of x would be less than the degree of b of x, okay, so you should be reminded of division with integers, okay when you see this, so you notice this polynomials over fx seem to be very similar like integers, similar to integers, right, so you have two integers, you can always divide one by the other and get a quotient and a reminder, okay, so you'll see the similarity is valid in more than one way, okay, so you can always think of polynomials in a similar way as you think of integers, okay, so at least as far as division is concerned, okay, many of the properties based on division carry over to polynomials as well, okay, so for instance you say if r of x is 0, what would you say, the reminder is 0, what do you say, say b of x divides a of x, okay, so a shorthand for that is just this, okay, so basically a of x will then be a multiple of b of x, right, it will be some polynomial times b of x, okay, b of x, so the way to read this is to say b of x divides a of x, okay, so that's the way to think about, okay, so there's also a notion of devices, okay, if you can find some polynomial b of x that divides a of x then b of x is supposed to be a divisor of a of x and based on that you can conclude whether or not devices exist for any polynomial, okay, so if you take any integer, the first thing people are interested in maybe interested in finding out is whether it's prime or not, does it have any devices, right, so one of those things which you learn a lot, I mean can you factorize an integer, so same way since division works, one can also imagine factorizing a polynomial, okay, but it'll be slightly different because see there's always, so there's always a trivial factorization of an integer, right, any integer can be written as 1 times n, okay, and you have to rule that out, you can't consider that as a proper factorization, similarly for polynomials there'll be more trivial factorizations, for instance if you want to see an example, okay, suppose I take a polynomial say a of x, let's say x squared plus 2x plus 1 and let's say it's in f5x, okay, suppose I say this, okay, suppose you want to think about factoring a of x, okay, so you want to write a of x as a product of two other polynomials, okay, there are various trivial ways of doing it, for instance I could write it as what, 1 times this, okay, that's too trivial, maybe 1 is not really something that you want, but you can also write it in a different way, I can write it as 2 times some other polynomial, what will I put here, I have to write it as 2 times, yeah, so you can divide this by 2, you can divide each coefficient by 2 or multiply each coefficient by 2 inverse, what's 2 inverse in f5, 3, right, so you multiply each coefficient by 3, you will get a polynomial there, so you would write 3x squared plus x plus 3, right, so is this a proper factorization, you have to rule out all these things, this is also trivial, okay, so this is also a trivial factorization, okay, so this is one thing I think it's true for any field, but I want to remind you that these factorizations are not considered factorizations, okay, so only what, which is considered a non-trivial factorization, if you can write it as, see each of these things should be polynomials of degree at least 1 and degree less than the total degree of a of x, okay, so that's only then it's a non-trivial factorization, right, each factor should have degree at least 1 and its degree of course will be less than the degree of a of x, okay, so that's non-trivial, okay, so can you find a non-trivial factorization for a of x here, x squared plus 2x plus 1, okay, so maybe from your knowledge of formulas for x plus 1 whole square you might be able to write it, but remember this is f5, will anything change now, won't change, you have to just reduce everything model of 5 and I've chosen numbers so that you're comfortably safe within that limit, okay, so you can see this will factor nicely as x plus 1 into x plus 1, okay, okay, so one thing you might have learned finding linear factors is very easy, okay, and for polynomials finding linear factors is typically very easy, okay, so why is that true, so this is how you find linear factors, finding linear factors of a of x is, okay, so how do you do that, if you want, if you have to look at that, there is a way of, there's a way of doing it, suppose, okay, suppose, suppose you have, so suppose let me say a of x belongs to fx, okay, so here's this result which will help you find linear factors, okay, a of alpha equals 0, okay, for some alpha in f, it'll take alpha in f, okay, if and only if, what, based on the division algorithm, x minus alpha divides a of x, okay, so a very familiar root based linear factor approach, you can easily, easily prove it using the division algorithm, right, right, so you put in a of alpha there, you'll see r of alpha will also be 0, okay, just work out very, very easily, okay, so you can show these things very, you know, very straightforward fashion, okay, so you can use this result quickly to find linear factors, okay, so even in a case where it may not be very obvious, the factorization may not be very obvious, right, the linear factorization, you can use this result to quickly find linear factors, so I'll throw a few polynomials at you now and ask you to find and ask you to try to factorize and the first attempted factorizing is to try to find linear factors, okay, finding higher degree factors is not as trivial as this, okay, so it can be a little bit more complicated, but linear factors can be very, very easily found, okay, so let's see a few examples to just try home this point, okay, so the first thing we'll see is the polynomial x square plus 1 belonging to f2x, okay, we'll see a very, very simple case, okay, how do you find linear factors, okay, you have to find roots for x square plus 1 in, let's say in f2, okay, I'm only trying to factorize it in f2x, okay, so you want to find factors for this in f2, okay, so f2 contains what, only two elements, 0 and 1, okay, the roots can be either 0 or 1, just substitute each of those values and see which one is the root, okay, so is x equals 0 a root, obviously not, okay, x equals 1 would be a root, so you know what, x minus 1 but x minus 1 is the same as what, x plus 1 in f2x, so x plus 1 has to divide x square plus 1, okay, so that is one thing we have found, okay, so you find that, okay, so if there is one linear factor for x square plus 1 what will be the other factor, it will also be a linear factor, right, in this case it will actually also be x plus 1, in fact x square plus 1 will be x plus 1, the whole square in f2x, okay, it's a very strange factorization, so you should immediately be reminded of, you should immediately convince yourself of one thing, okay, the same polynomial, if I change the field, the factorization will change, okay, so for instance instead of f2x if I had said this is qx, okay, will you have been able to factor it? No, in qx it doesn't factor, what about rx? Real number field rx, even there it doesn't factor, where will it factor? If you go to complex fields you can factor this, okay, so you can do that, so there is something there which is very interesting, so a polynomial that doesn't factor in a field can possibly factor in a larger field which contains this field, okay, so that can happen, yeah, okay, so that's something to watch out for, okay, just to drive home the point, let's try to see x square plus 1 the same polynomial, but I'll say now this is in f3x, so first thing you should convince yourself is x square plus 1 in f3x is a completely different polynomial from x square plus 1 in f2x, okay, the 1 is totally different, okay, the 1 that you had in f2x, if you add 1 to 1 what do you get? 0, in f3x the 1 you have, if you add 1 to 1 what do you get? 2, okay, it's not 0, and you know it's definitely not the additive identity in f3, okay, it's completely different, okay, so these two polynomials while I've written them down exactly the same way, they have the same only by looks, okay, by behavior they are completely different and you should not say these two are equal, okay, remember that, okay, but try to factor this, let's see, okay, you don't have any factors, okay, so it's fact is it is, the correct term is it's irreducible in f3x, okay, so if a polynomial has no factors you say it is irreducible, okay, so this factor is irreducible, this polynomial is irreducible in f3x, okay, so a polynomial which has no factors called irreducible and there is some distinction between irreducible and prime, in this case we can say it is prime, okay, that's not beat the point too much, but technically there is a distinction between irreducible things and prime things in theory, but we will just simply say it's irreducible, we just stick to irreducible, we won't say prime, okay, we just simply say irreducible, that's a convenient thing, we can think of it as prime numbers if it's confusing you too much, okay, this is a prime in that, okay, so you can keep looking at examples like this, it's quite intriguing to see the same polynomial factor in different ways in different fields, okay, so let's see one more example, okay, I went from 1 to 3, I think it's because I've read it f3 here, I'm sorry about that, I'll come back to my third, let's look at x squared plus x plus 1, I want you to look at this in f2x and also x squared plus x plus 1 and f3x, okay, let's see, try to do the factorization, in f2 it's irreducible, right, so you're able to quickly find it's irreducible, in f3 what happens? Yeah, it factors as x minus 1 times, see I have to be careful about minus in f3, right, because minus 1 is not the same as plus 1, okay, x plus, is that also x minus 1? Yeah, so it's actually x minus 1 squared, okay, so remind yourself that, okay, this is another reminder that strange things can happen when you change the field, okay, a polynomial that looks exactly the same will behave in a completely different, in one case it is irreducible, another case it has, no, don't, okay, so you can write it as x plus 2, but it's the same as x minus 1, it makes no difference, okay, so the integers, when you see them modulo something, you can write one integer modulo m in any ways you want, suppose you're looking at this modulo 3, modulo 3, 2 is the same as minus 1, it's the same as 5, it's the same as 8, right, I mean all these numbers are the same, once you say modulo 3, okay, so I like writing it as x minus 1 squared, okay, it's not the same as anything else, okay, so let's go to slightly more complicated examples, but now I'll just stick to f2x, okay, I'll just stick to f2x, okay, so let me start with x bar 3, I'm sorry, I wanted to write x bar 3 here for some reason, okay, from now on we'll stick to f2x, you can go to fpx, it's more confusing, just stick to f2x, okay, but we have to be slightly more careful when you look at higher degree polynomials, right, so for cubic polynomials, if it is reducible you will have a linear factor, I may write it wrong, you have to have a linear factor, right, you cannot escape a linear factor, if you have a factor of degree 2 what will be the other factor, it will be linear, okay, so anyway you should have a linear factor, okay, without having a linear factor this will not, this will not reduce, okay, so it's enough if you check for roots, okay, so you put in x equals 0, you don't get a root, x equals 1 you don't get a root, you can conclude this is irreducible, it's not a problem, okay, so let's try something more, something like say x bar 4 plus x plus 1, okay, so you can only say no linear factor now, you have to rule out degree 2 factors, I'm sorry, yeah, suppose it factors into 2, you can write it as a of x into b of x, what can be the degrees of a of x and b of x, yeah, so anyone has to have one degree, cannot have a linear factor, but this polynomial it's not true, it could have 2 factors both of degree 2, okay, so you have to rule out degree 2 factors as well, linear factors you can immediately rule out, right, you know immediately there's no linear factor, okay, so you put x equals 0 you don't get anything, x equals 1 you don't get anything, it's okay, but first for degree 2 factors you have to do some more work, okay, it turns out actually it's enough if you eliminate irreducible factors of degree 2, okay, right, so when you check for, check for whether a number is composite or not, integer is composite or not, what do you do, you only divide by primes, you don't have to divide by other composite numbers, right, if it's divisible by a composite number obviously it's divisible by a prime also, so it's enough if you divide by irreducible polynomials, okay, so you have to check for irreducible polynomials of degree 2, if you do that search you'll see this guy is the only irreducible polynomial of degree 2 and f2x, okay, look at all other degree 2 polynomials, all of them will be reducible, okay, if you want you can take a minute and enumerate all the degree 2 polynomials of in f2x, how many degree 2 polynomials will you have in f2x? Eight, right, eight, am I right? Okay, degree 2, man, degree 2 means one of the coefficients is 1, okay, the coefficient of x squared is 1, that's what I mean by degree 2, degree less than or equal to 2 is 8, you're right, if you say degree equal to 2, how many? 4, okay, that's correct, okay, you can write down all the 4 and you'll see this is the only irreducible polynomial, okay, all the others will have linear factors, okay, it's very easy to show that, okay, so it's enough if you check if x power 4 plus x plus 1 has x squared plus x plus 1 as a factor, please check that, how will you check it? You have to divide, that's all, you have to use long division, there's no other way here, you can't, right, you have to divide x power 4 plus x plus 1 by x squared plus x plus 1 in where, where should you divide that? In f2x, okay, do that long division and check if the reminder is 0, how do we know irreducible polynomials? Yeah, basically you have to list all the polynomials and check if it's irreducible, well, there are smarter ways and it's actually, okay, I'll comment about it as we go along, yeah, but for now, for starters, you can say I list all the polynomials and check whether each is irreducible, yeah, so it's, it's not a factor, right, you'll have a non-zero reminder, so this, this will end up being irreducible, okay, so you can do these checks if you want, okay, so it turns out there are quite a few irreducible polynomials, okay, it's not surprising and just like I said, there's an analogy between these polynomials and integers, right, these polynomials over a field behave very much like integers and there are quite a few primes as well, right, okay, how many primes are there roughly? There's some count, no, how do they grow in n? If you look at all the numbers from 0 to n, how many primes roughly can you expect for large n, log n, right, so that's the result, so log n is quite a large, when it grows with n, it doesn't, doesn't go off down to 0, right, so there are quite a few, so likewise here also, there are quite a few irreducible polynomials, so the typical way of finding irreducible polynomials, he asked a question is to randomly generate polynomials of a given degree, okay, and keep checking, you'll quickly find an irreducible polynomial, okay, so it's not very difficult in practice, it turns out one can show some very nice results for every m in n, okay, positive integer m, there is a, there is at least one, in fact, typically more than one, at least one irreducible polynomial of degree m in, I'll say fpx now, this is what we need, p prime, any p prime, this is the result that we need now, in fact, you can generalize this a little bit and show any finite field also, it will be true, okay, so we know there is for every degree m in every characteristic p, okay, in every fpx, I'm sorry, I haven't said anything about characteristic items, sorry for that, in fpx, there will be at least one irreducible polynomial, okay, so that's the result we'll accept without proof, okay, if I have to prove it, it's quite unnecessary, it's a tangent, so we'll accept this without proof, okay, so that is just like we accept that our prime numbers, we'll accept this also, okay, so this is true, okay, any questions about irreducible polynomials, okay, these will play a very important role in the construction, so if there's something disturbing me, disturbing you about this, you should ask me right now, is there any question, okay, all right, good, all right, so we've seen enough facts, I think about polynomials and this is pretty much what you should need, okay, so I don't think we'll need anything more, so we'll, let's try to jump into construction of finite fields, okay, so we'll do it, we'll do this in a very simple fashion and in many ways this, the way I do it will not be very rigorous as well, okay, so if you're interested you should read books and a good book, very good book is Algebra by Michael Aten, a cheap edition is available, strongly suggest, very strongly suggest that you are interested in these things, please buy this book for yourself and go ahead and read it, it's a very good book and if you want to really understand all the technical details you can look at it, okay, so I'll do it in a way which is simple, simple and intuitive and appeals to all, okay, it's not a very rigorous way, okay, so first thing we'll see is we'll assume, okay, so far I've not shown you anything, only finite fields we've seen so far are what, fp, okay, so we've seen fp, p for p prime and hopefully you're reasonably familiar with this, okay, so one of the results that I showed for groups which will be very crucial later on also as we go along is order of an element, okay, so order of an element in a finite group, what does it do, it divides something, what does it divide, divides the size of the group, okay, so if I take for instance fp, fp star is what, it's a multiplicative group, multiplicative order of any element in fp, okay, we'll divide what, p minus 1, okay, so be very careful, if it has to divide p then you'll be happy, because very few things divide p, okay, so it has to divide p minus 1, okay, so just a reminder so that you don't forget that result, okay, so now we will try to prove some properties assuming fq exists for some q, okay, I have not said anything about existence of fields, I'll say, suppose, no, I didn't say such things, I said any two fields of the same size, there'll be isomorphic, that is the result, I never said the size is only p, and I'll come to it, there is a close relationship between fp for a prime p and any field that you can possibly have, I'll come to it, there's a very close relationship but it's not definitely not isomorphic, there are other fields than fp, fields other than fp, okay, suppose fq exists for some q, okay, some very powerful results are possible for q, okay, just based on simple arguments, you can show some really, really powerful results for what q can be, okay, some of those surprising results, if you haven't seen it before it'll be very surprising for you, okay, so suppose fq exists, okay, I don't know anything about fq, I'm just saying fq exists, okay, what all should I define for defining fq? I should define a set with q elements, okay, I should define, I define it such that fq with an addition operator becomes a group, okay, and then I should also define it so that the fq star which is the additive's identity thrown away should be a multiplicative group, okay, I'm not saying anything about all those things, okay, I'm just saying it, I assume it exists, okay, I don't know about the elements of fq but I know two elements of fq should always be there, what are the two elements that I know should be there? The additive identity of the additive group fq and the multiplicative identity of the multiplicative group fq star should be there, okay, 0 and 1 belong to fq, I know these two will be there, okay, so one might start by saying what is 0 equals 1, okay, so if you multiplicate the identity and additive entity are the same, you don't get anything interesting, okay, so it's just everything is 0, okay, you don't get anything, okay, so we'll obviously assume 0 and 1 are not the same, okay, so what's the only way of coming up with other elements, okay, I know there are operations that are plus and dot, those two operations have not defined but they exist in this field, if the field exists they have to be there, using those operations, can I create more elements of fq just using 0 and 1, okay, right, you see that's just about the only way I can create, the only way I can create more elements is if I add 1 to itself, right, if I do anything else I won't get new elements, okay, right, so I could think of 1 plus 1, okay, this will belong to fq, okay, remember I don't know what plus is, I'm just saying there should be a plus and I can always add 1 to itself and get another element in fq, there's nothing that stops me from repeating this, I can add 1 plus, 1 plus 1 and I'll still be in fq, okay, so now since I can keep on doing it I'll need a shortcut, shorthand, okay, so instead of writing 1 plus 1 plus 1 n times, okay, what will be my shorthand, okay, remember I'll write this simply as n, okay, I'll write this simply as n itself, okay, let's assume that it's n times 1, okay or n, n, 1 added to itself n times, it's what it's assumed to be, okay, n, okay, now I know q is finite, okay, so remember q is finite, right, okay, so but this n can keep on going, right, it can keep on adding and obviously since fq itself is finite, since all these things have to belong to fq, eventually what should happen? Yeah, you should go to 0 itself, okay, in fact it has to repeat but from that argument you can easily show that there will be a smallest n for which n will be equal to this 0, the additive identity is 0, okay, do you notice that, okay, so since q is finite, there exists smallest n belonging to n such that n equals 0 in what, in fq, okay, this n that I defined in fq, how did I define n, how did I define n in fq, I took the 1 and added to itself n times, that's how I got the n in fq, okay, this n in natural number is little different, right, that's just the number, okay, there will exist a smallest n such that n is 0 in fq, okay, the next surprising result is, okay, this smallest n has to be prime, has to be a prime number, okay, okay, so how will you argue that it has to be? Okay, that all those things need a lot of proofs, okay, I haven't come to that, eventually I'll show some isomorphism like that but let's not go to that but why should, what's the simple argument for why? Exactly, see, so why are you going to fp now? There's no going to fp, there's no field, you know, I mean just let's not jump into something else, eventually we'll get that, the simple argument is suppose n were not prime, I can write it as a times b and even in fq that will hold, right, okay, so how do you prove it? If n is not prime, okay, if n is not prime, if n equals ab in n then n will be a times b in fq as well, okay, why will that be true? You can, you have to write that and prove it by the distribution properties, you can prove these things, okay, you can take the distribution property of the field, you can take the distributive property of the field and you can show a times b will be n in fq, okay, it's very easy. You remember, how did I define a in fq?  You can use the distributive property of the field, it's just one, just keep on doing it, you'll have same one added to itself n times, so we'll get the n, okay, now if n is 0, what have I shown? a or b has to be 0 in the field, okay, so then what will be violated? n will not be the smallest entry anymore, so obviously there should have been a smaller number, okay, so therefore n has to be prime, okay, and 0 equals a b implies a is 0 or b is 0 and you get a contradiction, okay, so therefore n has to be prime, okay, so this prime number which is the smallest number for which 1 plus 1 plus 1 added that prime number of times equals 0 in fq is called the characteristic of fq, okay, so this is the definition of characteristic, smallest prime p for which p equals 0 in fq, okay, remember what is p in fq? 1 plus 1 plus 1 added b times, okay, so remember this is a very powerful result, okay, so I never knew, I don't even know if fq exists, okay, I only know, so if somebody tells me that fq exists, then there has to be a characteristic for this fq which is p, right, it's a prime number, which is a prime number p and that p will be 0 in fq or 1 added to itself, p times will be 0 in fq, okay, so what have I shown right now? So just based on this, I've shown 0, 1, 2, 3, all these till p minus 1 will be actually a subset of fq, this is what I've shown, okay, for some p prime, okay, p is prime, okay, I will call this set as fp, okay, remember I have not shown that this is the field fp, okay, I'm going to call this set as fp, right now this is some subset of fq, okay, and all the operations among these elements are defined in fq in some field which I don't even know whether it exists or not, okay, but the only thing I know is it's got p elements and each of the elements are 1 added to itself that many times, if I say 2, it's 1 added to itself twice, if I say 3, it is 1 added to itself 3 times in fq, okay, of course since I've called it fq, what am I going to show? I'm going to show next that this set, this fp will be exactly the fp that we saw before, okay, it'll be isomorphic technically but we'll simply take it as equality, okay, okay, so that's the next result we'll see, see fp contained in fq is actually the finite field fp, okay, so how do you show these things? How do you show that this set will be the same as the finite field fp? You have to show, yeah, exactly, you have to show every operation here will result in the exact same operation in the original fp that you knew, it has to, right, you take any two things and add, what are you doing? You're taking one and adding to itself, so it'll be the same thing as the sum and why do you have to do modulo p? Where will the modulo p come? Because p is 0 in my fq, I've shown that, okay, so modulo p will naturally come because p is 0, whether you multiply or add, multiplication is also easy because then use the distributive property, okay, so it's all just sums of ones, so you can show a times b is same as a times b, where will the modulo p come? Because p is 0 in fq, okay, so the exact same thing will come, okay, so you can write down the proof, this requires proof, okay, so based on your familiarity with mathematics, it might be easy or difficult for you but it's very simple proof, there is no fancy concept here, okay, so imagine what we have shown now, it's quite powerful, I've said fq is some field, I don't know if it exists or not, but just because it exists, what should it contain? It should contain some fb, that's the only thing that we have shown so far, okay, is that clear? Any questions? Any thoughts on what is it that I've actually shown and what is it that I'm claiming? Oh, it's okay, it's clear enough, okay, okay, so the next statement we'll show will be slightly more, yeah one of those to see because if one of them are non-zero, suppose a is non-zero, it'll have an inverse, multiplicative inverse, so multiply both sides, you'll get b to be 0, so any one has to be 0, but that can be shown, okay, so there are some the next result is even more interesting, okay, so now any, so you have a bigger set fq, which contains the field fp, okay, and what can you do in fq? You can add two elements of fq, and it contains a field fp and you can also multiply two elements of fq, okay, so naturally you'll get addition and scalar multiplication and fq will become a vector space over fq, over fp, okay, so I don't know if you're not familiar with abstract vector spaces, this might be a very surprising fact to you, but what is how do you define a vector space? What's the definition of a vector space? It's a set of vectors over some scalar field, okay, so you need a scalar field and then you need a set, what are the, what are the, what else do you need for a vector space? You need to be able to add any two vectors and result in another vector, can you do that in fq? Can you add two elements of fq and result in another element of fq? Yeah, that's what the addition operator will give you, okay, and now my base field is fp, can I multiply an element of fp with an element of fq? Yeah, I just used the multiplication operation that was defined in fq, I can multiply, I'll still be in fq, so both of those hold and fp is a field, I know it's a field, so fq will have to be a vector space over fp. Sir, can we tell that any field is a vector space over itself, yeah, if you take a field, yeah, it will do, right, so you notice the argument, okay, so it's a slightly abstract argument, but basically the way you argue is if you have a and b belonging to fq, a plus b belongs to fq, that's my vector addition now, this is addition in fq, which I will call as vector addition if you want, okay, what about multiplication? If you have a belonging to fp and b belonging to fq, how can I do a, a dot b? Yeah, it will also begin in fq, this will be multiplication in in fq, I will call it my scalar multiplication, okay, I can happily do this, okay, so fq will have to be a vector space over fp, okay, now what do I know about vector spaces? Once I come to vector spaces, I mean very familiar domain, okay, fq is in fact finite, it's only got a finite number of elements, so it's very easy to see, fq will in fact be a finite dimensional vector space over fq, fp, okay, so that's the next step, so more is true, fq will be a finite dimensional vector space over fp, okay, what else is true? Suppose I say, suppose dimension is m, okay, it has to be some finite number m, I will say dimension is m, okay, now what else do I know about vector spaces? What else do vector spaces have? How do I describe all the vectors in a vector space? Use a basis, okay, so that implies now there exists a basis, let me call it b1, b2, how many vectors will be there in the basis? bm for fq over fp, what does it mean now? An arbitrary vector in fq, let me call it alpha, I'm just sorry, x, x, what else? Alpha and fq can be written as alpha equals a1, b1 plus a2, b2, where will all these gates belong? ba will belong to what? Fq, okay, it's the basis but there will be elements of fq, right, all basis vectors are belong to the vector space, right, so we all belong to a2, b2 plus am, bm, so where those basis, where do ai belong? fp, okay, for every alpha and fq, I can uniquely write it in this form and it's also vice versa, for every set of ai that I come up with, I will have an element in fq, so how many elements will there be in fq? How many choices do I have for each ai? p power m, so each ai can be an arbitrary element of fp, right, so that would be p possibilities, I have m of them to choose, so q has to be equal to p power m, okay, so I wanted to step back and imagine the power of abstract mathematics at work, okay, so started with fq, you don't even know if it exists or not, okay, said somebody tells you fq exists, the two pages of work, you can come to the fact that this q has to be equal to power of a prime, it cannot be 10 for instance, okay, so that's an amazing thing that we have shown, okay, so it's quite non-trivial, okay, you should be surprised by it if you're not surprised, okay, so just because somebody said a finite field exists for q, you can show q has to be equal to p power m and the proof is very, I mean it doesn't involve any magic, right, it's some very simple step-by-step argument, there's nothing more to it, I don't think I've really missed out on any important step in the proof, I've shown pretty much everything, okay, so it's quite simple to see, q will be p power m, okay, so now this analogy is more powerful than just saying q, just giving you an idea of what q can be, okay, what else have I done, how did I go to the vector space, what was my addition in the vector space, it's the exact same addition as in Fq, so what will be addition in Fq, it will be addition of vectors over Fp, so how do I add two vectors now, okay, addition in Fq, suppose I ask you to add two vectors, alpha 1 and alpha 2 in Fq, if you have to add it, what will you do first, first write it as a vector over Fp in some basis, okay, we'll choose some basis and write it over Fp, okay, so you write alpha 1 as a11 b1 plus a12 b2, so until a1 m, bm, and then you write alpha 2 as a21 b1, a22 b2 plus a2 mbm, then what is alpha 1 plus alpha 2, it's very easy, this is simple vector addition, a11 plus a21, why do I know how to add a11 and a21, both of them are in Fp, simply add mod p, there is nothing more to do, okay, b1 plus so on till a1 m plus a2 mbm, in fact, we have a representation for each element of Fq, okay, we don't know it, we didn't, do you see that it, Fq is just an m dimensional vector space over Fp, how will you represent vectors, m dimensional vector space over a field, you can use coordinate vectors, okay, suppose, okay, only thing I don't know is I don't know the basis, right, I haven't told you what the basis is, I don't know the basis, but I can represent every vector as simply a coordinate vector, I just won't be able to translate it into the actual vector, but I will know what the coordinates are, right, just list out all the coordinates, there are p by m possibilities, right, simply write it down, okay, and you can even add them, okay, only thing is you won't know what they actually are, okay, you will only know their coordinates, okay, so in your, in the vector spaces you are used to, like R3 and all, coordinate is everything, right, you don't care beyond that, but this vector space coordinate is not everything, because you don't, you don't know the real elements, the b1 and bm, you don't know it, you don't know it, actually it will turn out also very simple, we can find that very easily, but we don't know them yet, but we can represent each element of Fq as coordinates, and in fact add them also, there's no problem, just add the coordinates, individual coordinates are added, modulo p, okay, so let's see that example right away, okay, suppose I say F16, okay, I haven't told you if it exists or not, suppose somebody tells you F16 exists, okay, then I can write F16 as what, I can write it as<PHONE_NUMBER>0 0011, so on till what, 1111, okay, and I even know how to add, how will I add 0011 and 1111, simple vector addition modulo 2, that's all, for instance 0011 plus 1111 will be what, 1100, I know it belongs there, what is it that I cannot do in a field, what else should I be able to do, should be able to multiply and I have no clue how to multiply, because vector space will not take you anywhere as far as multiplication is concerned, right, multiplication, how did I argue for multiplication I said, I have to use the multiplication in Fq, I mean I haven't defined what multiplication is, that's the only other trick we have to add to this construction, okay, so we already know what F2 power m or Fp power m will look like, it'll be all coordinate vectors over Fp and I also know how to add them, okay, the addition is also going to be the same as vector addition, the only thing I don't know is how to multiply two elements of Fq, okay, you don't think of multiplying two vectors, but I mean it's the same, how do you multiply two elements of Fq, how do you multiply these two, to any two vectors in this, okay, so hopefully you're convinced by now that finite fields are the simplest things in the world, yeah F16 will have to be over F2, no, right, what will be the characteristic of F16, has to be 2, okay, so right, why does it have to be 2, q has to be p power m where p is the characteristic, okay, so maybe I didn't write it down, Fq exists implies q is p power m where p is characteristic, okay, any q can be uniquely written as p power m, if it has only one prime factor it'll be very easily figured out, so once I give you q you can find out what the characteristic will be also, okay, so there's no F16 over F3, it's nothing like that, only for vector spaces you have, since I've said it's already a field that is a more complicated multiplication, once you define that you'll see, I mean it cannot be over arbitrary fields, it doesn't work, okay, so any questions on how this worked out, okay, all right, okay, so good, so the next thing we have to show is multiplication, okay, and for multiplication I will use, I will basically what I'll do is I'll present the construction for Fq for any q which is p power m and then I'll argue why the multiplication works, okay, but I want you to think back over Fp, okay, if you had worried only about addition, Fn should have worked for any n, okay, multiplication was made possible why, we're doing modulo p and p was prime, okay, so we'll use a similar idea here, we'll construct a set of objects, modulo some other thing for which will make multiplication work, okay, so basically that's the next thing we'll do, I'm dangerously close to the end of the period, I think we have 10 more minutes, I think I have time for doing this, okay, so let's do construction of Fq, okay, I think I already wrote down, construction of Fq for q equals p power m, p prime, okay, I know all these things now, previously I didn't know, I know all these things have to be true, Fq cannot exist, just by the same proof you can show that Fq cannot exist, okay, I assumed, I started with the assumption that Fq exists and then I showed q is p power m, so if q is not p power m obviously that cannot exist now, anytime it exists you have to come to this q equals p power, okay, all right, so this is how you construct it, I'll simply present the construction and then I'll go back and appeal to this as well, okay, so you can show Fq which is defined as a of x in Fpx such that set of all polynomials in Fpx such that degree of a of x is less than or equal to m minus 1 can be made into a, can be made into a field, okay, okay, first of all some sanity check, okay, first of all does the set have p power m elements, if I say set of all polynomials in Fpx degree is less than or equal to m minus 1, it will have p power m elements, okay, that's the first check, based on what I know about addition in Fp, Fq, does that same thing work, what if I add two polynomials, what will I get, is it the same as vector space addition, yeah, it's the same as vector space addition over the coefficient because these x powers are just placeholders, they're just telling you what to add and it's exactly like vector addition, so both of those work, okay, so whatever I know about Fq is not contradicted by the set it and the polynomial addition operation, so I can make the addition a simple polynomial addition, okay, okay, so for multiplication we need we need something prime, right, how did multiplication work the last time, you did it modulo, some prime number, here we will need an irreducible polynomial of degree m, okay, so I'll start with pi of x being an irreducible polynomial, irreducible in Fpx degree equal to m, okay, why do I know such a polynomial will exist, yeah, I just gave you the result, okay, so it's true that for any m, any degree m I'll have an irreducible polynomial pi of x, I will define multiplication modulo, polynomial multiplication modulo p of x and pi of x, okay, so what do I mean by modulo in polynomial things, when I multiply two polynomials and if the degree is less than or equal to m minus 1, obviously the degree can be greater than, right, so if I take two polynomials from this field and multiply I can go outside, how do I reduce it, I'm going to say modulo pi of x, what should I do when I do modulo pi of x, I should divide by pi of x and take the reminder and the reminder will definitely have degree less than or equal to m minus 1 and it will belong here, okay, so this multiplication satisfies the closure property, okay, you multiply any two, you will definitely come back to this field, there's no problem, okay, but what's the crucial thing you have to check in finite field, in field multiplication, inverse and inverse will exist by the same argument that I gave before, okay, you simply take a polynomial and multiply by all other polynomials modulo pi of x, no two of them will be the same, why, if any of them are the same pi of x will have to be reducible, the same argument is before, okay, since that cannot happen no two of them have to be the same, since all of them are distinct and you have exactly q of them, one of them will have to be equal to 1 and that will be the inverse, okay, so there's no change in the proof, okay, so that's why I kept saying these polynomials are the same as integers in many ways, okay, whatever you did with integers you can do with these polynomials also, you'll have these prime numbers as far as factorization is concerned, they're very very similar, okay, so that's the proof, I don't even have to write it down, right, all of you are convinced you're all shaking your head very vigorously, okay, right, so it's a very easy proof, the same proof will carry over as an Fp, okay, just pay the same proof this will work out, okay, so that's it, that's the end of the construction, these four lines, okay, who were told you finite fields are difficult, I don't know what they were saying, right, they're the simplest things in the world, okay, how do you construct any finite fields, simply take all polynomials of degree less than or equal to m minus 1, how do you add simple polynomial addition, how do you multiply, you find the reducible, find some irreducible polynomial of degree m, you do model work, that's it, exactly, that can be a basis, so now we can go back and find basis and anything else if you want, okay, so I know what the objects are, I know what the vectors are, okay, so I can easily go back and find basis elements and all, I'll come back to that more, okay, all right, so we'll stop here, do more


Welcome everyone to our bill analysis workshop, part of a series of workshops to help the public our members and supporters better understand the legislative process. So last time we did a general overview of the process. And today we're going to dive a little deeper into specific legislation to better understand how it works. And I'm going to turn it over to Bob how our lobbyist and he presenter tonight to lead us through this workshop. All right, well, hello everybody my apologize for the delay on my end. So, I have represented the Sierra Club now for three or four years and representing clients at the state house for 40 years. Right of that I served in the main house of representatives. And I live in the town of Brunswick. Now, Matt, do you want to allow me to share my screen. Yeah, you should be able to your coho so you should see that screen. So tell me. The first thing I are you seeing my what I'm seeing on my screen which is the legislative reports here club. Yeah. Okay. Is it. Is there something I have to do to accept. No, you are a cohost so you should be able to share. Voila. There we go. So, what you're looking at is the report I put together for the Sierra Club for bills I think Sierra Club is interested in and we add to this or maybe it takes something off. As I meet with Matt and the legislative team to discuss what issues Sierra Club wants to follow and and indeed get involved in. So, why don't we jump right in the first these are bills that have been posted, that is to say the fully drafted they've been assigned LD that is legislative document numbers. And we can see exactly what they do in most cases. Anybody is welcome to get this link. I have an email that goes out automatically to clients on Fridays for the link to the clients report. If you're not getting those and you want them let me or Matt know and I'll add you to the distribution list. You mentioned that the interest is 5050 and whether to take a look at the plastic bag band or was it. I was so involved trying to get my technology straight here I didn't catch everything you said. Yeah, that was for the bag band or offshore wind but maybe we could start with the hydro fluorocarbon and then I want to be the vice president and break that tie I'm happy to delegate that to you. Maybe somebody else will come on and take that responsibility. What bill number am I looking for here to the number 226. Thank you just pass it. All right, let me let me go up just a little. Here we go. All right. So this is the LD number if you can see my cursor circling around this is the bills title and you can see that that's a hyperlink. This is the description of the bill taken directly from the bill. Over on the over on the right here that's simply when I last made a change to this entry. The status is it's simply pending final legislative action because there hasn't been much action. This is the name of the committee of jurisdiction. We do not have a hearing or work session date. Let's look at the bill itself on the legislature's website. We do that by going here, clicking on that hyperlink. So this is a page from what's called the bill directory. And there's quite a bit of information on here. If we want to look, see what a legislative document actually looks like is here it is in PDF form. This is not at the front page is always the same state seal. We're at the beginning of the 130th main legislature, which will serve for two years, and then there's an election and then we'll be going to the hundred and for 31st main legislature in 2023. This is a document number. This one. First was reported to the House of Representatives and it went there first because it's primary sponsor is a representative. This line right here submitted by the Department of environmental protection would appear on any bill submitted by a department. This is represented by a legislator and Ralph Tucker, who happens to be my representative submitted on the behalf of the DEP because he's the house chair of the committee that will hear the bill. And you see here that the clerk of the house has referred the bill to the committee on environment natural resources, which Ralph chairs. This is the substance of the bill. A lot of this will look like some sort of foreign language perhaps to you, but it is adding a definition to some existing law. But probably the best way to start is to go straight to the end of the bill where there is a summary, and then you get a better idea. What the bill does. We're going to go straight down until we get to the summary and I went right past it. This is it. And this is what the bill does it prohibits the sale lease rent installation use or entering into commerce of any product or equipment that uses or will use a substance that is a hydro floor carbon with high global warming potential intended for any air conditioning anywhere. So propellant end use as determined by the DEP tracks the DEP to adopt rules. And in adopting the initial rules. The department must regulate each substance and end use as specifically provided for in the bill and may not regulate any substance or end use, not addressing the bill. The department may adopt rules adding or removing substances from the list of prohibited substances or adding or removing and uses. Well, what is all that mean. I think it's fairly straightforward. What this covers hydrocarbons. With certain intended uses the rulemaking stuff. That's pretty common, particularly with legislation that deals with it. Highly technical or complex subject. Rather than the legislature, the citizen legislator trying to figure out exactly. How the rules should be written. They delegate that task to the department. Now, they can the legislature. Tell the department before the rules are implemented that they have to come back to the legislature for review. They do that if they deem the rules to be major substantive rules. And perhaps we can find up here whether they've done that rulemaking. No, they did not do that. The department should adopt rules to implement the section rules adopted by our routine technical rules. So there are two types of rules are major substantive rules which come back to the legislature review and approval. And then there are routine technical rules that simply go into effect once the department has conducted the rulemaking process. Which is prescribed in law. So, let me, before I go any further, let's see if there are any questions or Matt, whether you want me to steer this in a particular direction from here. Feel free to. Yeah, if there are questions, you can put them in the chat. I don't see any at the moment, but. I appreciate I appreciate knowing the distinction between major and routine technical major substantive. It's strictly a call by the legislature which type they are going to deem a set of rules to be my guess is they've done routine technical rules here because there's still there's already a very well developed set of rules on on the regulation of chemicals and substances and this simply adds another category to it. It's not it's not a whole new area of law. So what else strikes you about this legislation was this also was this given by the department or the governor. If this comes from the department. It's not a governor's bill per se. The governor can put in her own bills and she can do that anytime she wants. Whereas departments. Had a deadline in I believe early December. And then individual legislators submitting bills. In their own right and not behalf of the department or agency at a somewhat later deadline 18th of December. Now I will say the department is very unlikely to put a bill in without checking with the governor's office to make sure the governor is. Is okay with it. But it is not per se a governor's bill. I think mine had his hand up. No, you're batting away flies. Any other questions. Well I guess this does bring up what Bruce briefly touched on but the the rulemaking process so I mean this is legislation to change. And then with specific definitions by the department and then there, there is a rule there rulemaking processes for most agencies. Can you just give us a clear try to understand the difference between rulemaking and the legislative process. In this regard, sure. Well, let's let's start with the foundational governing document for the state of Maine and that's the state constitution. Everything flows from that. The legislature cannot pass laws that are somehow in conflict with the state constitution. Likewise, the department or agency cannot enact rules that are not consistent with the statutes enacted by the legislature. But assuming this bill. Well, there's no question about what their rules are likely to be consistent with the intent of the legislature because they're being directed by the legislature to do this. There is a quite a body of law statutory law governing the rulemaking process it's called the administrators administrative procedures act we could. You know, I could bring that up on the screen if anybody wants to get into it, but maybe this isn't the session for that. Suffice it to say perhaps that the public has to be given notice of proposed rulemaking. There is often a public hearing. I believe in all cases there is the opportunity to submit written comments. And then only. And I believe that the agency has to respond and writing to each set of comments. And then eventually they can promulgate the rule. And it has the force of law. The difference. If this were deemed a major substantive rule. Would be that after they've gone through that process they age of the D people not promulgate it until they send it to the legislature. And it essentially comes in like a bill. And the legislature can approve the rule. They can approve it with changes or they can reject it. Pretty much the same. Options that apply to any piece of legislation. And all all those steps and great detail, except for within the administrative procedures act. And that applies to any agency. That's. Has rulemaking authority. And agencies needs specific authority to adopt rules. If the legislature hasn't authorized an agency to adopt any rules, they don't adopt any rules. Okay. That's helpful. And I guess my only other question, I think I'm going to send another quick poll out in terms of how many bills people want to cover. They want to go more in depth on this one or cover more bills, but I guess my only other question before that would be, what, what do you think of this legislation does this seem. When does it seem likely to pass. What are your impressions. Well, the substance of what this would be a useful exercise. There's a little preamble up here be an active people, the state of Maine, as follows. This is a section number of the bill. It pertains to this 38 here is a title number. The environmental laws are entitled 38 there are 40 some I think titles, titles, title numbers, number titles there. MRSA may means main revised statutes annotated. Which is our code of statutory law. Bill squiggly is the symbol for a section, and it's creating a new section entitled hydro fluorocarbon use restrictions. It defines the term substance. I guess what I'm looking for. They don't. They don't in the bill to find hydro fluorocarbon it may be addressed in existing law. Then there's a prohibition regarding these substances, and then there's a rulemaking authority. So everything from line one to line 15 is new law. Everything else from there on and you can tell that because it's underlined. The rulemaking authority is not underlined. I mean to make folks dizzy here. The rulemaking authority is not underlined because it's existing law, I believe, and we can check that by going. And Bob, as you do that I'm going to send out a poll because I mentioned earlier that this is kind of like choose your own adventure we're trying to gauge your interest in what you all like to get out of this so I'm just going to throw out a poll kind of wondering if we want to just focus on a couple bills. Pretty thoroughly like we are now, or kind of roll through a few more. And focus kind of just on the general overview content. So I'm just going to launch this see what people think. But carry on Bob. Yep. So, um, I'm assuming you can see something that says title 38 chapter 16 sale of consumer products affecting the environment. And these are all existing laws. Parasol sprays chemical septic tank cleaners phone products lead acid batteries. Plastic bad bags adult I think now elsewhere in law motor vehicle air conditioning wheel weights lead in them used to be ozone cleaning products bromide flame retardants posted furniture electronic way. So I'm just going to move to another area of the lawn. Oh yes his plastic bag reduction. So, see that that last one is section 1611. The new one is 1612. So when this is enacted. You go to this page and you'll see one other line here below this one. But if we wanted to look at plastic bag reduction, we click there. And it will take us to the current along plastic bag reduction. Just looking down to see if that contains its own rulemaking section. I'll go too far down the rabbit hole. You asked me if this is well drafted. I think it's very well drafted in that is. Detailed to the nth degree is really not much question about what it does. It's a little different than a lot of bills because of all of this stuff in here that deals with existing types of things that are regulated by the dp. As some degree of hazard. I'm guessing that might well be enough on this bill, but you tell me. Yes. I think so we got our poll result. Well, if you haven't voted. You have five seconds. But we have definitely the majority of people have voted. And yeah, and just before I show the poll. Isabel our legislative team regularly dives in deeper and I think our next workshop. I have to double check but I think we might. We might also look a little bit deeper as well. So once we do a few of these, it'll make more sense and you even just reading the summary really helps. But it looks like our polls with about two thirds reporting are to kind of move similar to this pace and focus on the few that we first talked about so this one and to maybe maybe three more if we can but the next one can be the pine tree amendment and then if we have some time we could just do both of those opposition bills that the band and the offshore wind. So you're talking about the environmental constitutional amendment to establish environmental rights. Yes, I think it's 60. It was 64 or five on there. That I have maybe I have it up at the top in the priority section. Yes, I do 64. Yeah. Yeah. Okay. This will be a little different because it proposes to amend the state's constitution. So here you have the LD number. This is a title. We've description. It's going to. Well, it's going to the energy utilities and technology committee that's interesting, although not terribly surprising. I would have thought, however, we go to environmental environment natural resources, let's click on the link. And we'll open the document itself. So you've seen the format title. Oh, now I entered. I entered the committee on energy utilities, which was obviously an era. It's what I thought would have thought it was so I'll go back and change it. Sponsor. So it adds a new provision of the state's constitution and article 10 section 26. It establishes certain environmental rights, which are set forth here. And then. There's language that's unique to constitutional referenda that spells out the process. Including how the question will appear. On the ballot right here. Now, the The only way the main constitutional can be amended is if the legislature sends a question out to the people. Constitutional amendments cannot be initiated by the voters like statutory law can be. So, for example, we've seen. Citizen positions, petitions on the CMP corridor and the returnable container law and any number of other things, but you won't see a citizen mission petition. On a constitutional change access to start in the legislature. And not only that, it's not easy to do this and attentional. The legislature has to do this by a super majority of two thirds of the house two thirds of the Senate in order to send it to the voters. Like a lot of constitutional language. It's, it doesn't have a lot of detail. If this passes no doubt, you will then see pieces of legislation. Responding to this perhaps to create some new. Public hearing rights that the constitution guarantees or how to implement those rights. Not been scheduled for public hearing yet. As far as I know, we can go back to that page. And if one were to click on status and committee. No public hearings or work sessions have been listed for this bill. So these other links over here won't give us anything yet because the bill hasn't been heard. No amendments been offered. No committee divided reports. Nothing's happened in the chambers yet that is to say that full house and Senate. The actions that have been taken. That's it so far. Refer to the committee. And I'll go in and change the committee that I listed on my report. I made a mistake. I don't know. Any questions on. On LB 64. I don't see any yet, but just to clarify. So this would need a two thirds vote of the legislature. And that's it and a majority or two thirds vote of on the ballot. Majority of the voters, but. Majority of the voters. Okay. Yeah, I don't see any others. I don't see any. I don't think to. For a way to keep track of this. Specific campaign. And I did mention that. I believe there's another bill relating to this that. Is. Better in terms of what are. Our folks are interested in supporting. It's very similar, but. It's very similar. It's very similar. It's very similar to the competing bills of a similar topic, even for ones that amend the constitution. And somehow. Because there's no, there are no limits on the topic or number of bills that can be put in in the first year of the two year. In fact, there are some exactly identical bills that have been submitted. In the normal year, which this isn't. In fact, I would have those two identical bills combined into a single one. But COVID's changed the way a lot of things are done. And this office I refer to is called the office of the advisor of statutes. They haven't been trying to. Combine identical bills. They're letting the committees. Sort that out. One committee. Had hearings today on two identical bills. But they'll report. One of them out of committee eventually. Yeah, I'm not sure off the top of my head, which. The other. You're talking about another proposed amendment to the constitution. Yeah, and I'm not, I'm not quite sure how. I guess it just hasn't been printed yet, but it was introduced, I think as an LR. Yeah. That's another thing we might mention briefly. I'm not sure. I'm not sure. I'm not sure. On December 18. The deadline for. Individual legislative file of bills. They're about 1600. Had been filed. And one can go and look at a list of the titles of those bills. They won't all become printed documents. And. And there's no. The only thing that shows up as a printed bill, it really depends upon when the revised office gets to it. But that process. Bill will, will continue to emerge. Over a period of two or three months. Maybe even four. Once they posted like. Like, well, let me go. Let me do this again. Once they're posted in this format. We'll have a rules call for two week public. Notice of the hearing. And one can get on a list. Each committee has a distribution email list for notifications of its activities, including public hearings. And one can get on that list. Perhaps Matt, before we're done, I could do a screen share of the legislature's website and just. Point out a few things that. I think are very helpful in trying to get information about what's going on under the capital don't. Yeah. Whenever you think it sounds like maybe at the end, but I was just going to suggest maybe as a segue, you were just on that plastic bag band. Maybe. We just hop into that and it would be great. It seems like timing is, is going okay. But feel free to ask questions. Folks. I think we still could cover that offshore wind. Yeah. I think that's a good idea. I'm going to try to repeal as well. Cause I actually haven't read it yet either. And I'd like to look at it. But we just looked at a constitutional amendment. We looked at. Department bill that. Defined. Hydrofloor carbons and what should not be allowed. And now. We have a bill that is trying to. We have a bill that was passed last time. Yeah. And I'm trying to get us to. The law. It would repeal. Well. There with me. And so you're 16. 1611 is the one I want. Just. 1611. Which is right here. So this is the current law. This is what the bill would repeal. So. So. First, you've got. Some definitions, including. Plastic. Post consumer recycle material. So forth and so on. And then you've got a list of. They define restaurant retail establishment. What's a reusable bag. What's a single use carryout bag. And then after the definitions. And then exemptions from the prohibition. So. This sentence here contains the prohibition. And then following that under B, a bunch of exceptions. Retail establishment may not provide a single use carryout bag to a customer. At the point of sale or otherwise make single use carryout bags available to customers. But then. Here's a list of exceptions to that. My favorite newspaper bags. We use them for doggy poop bags, but that's all right. At least they get second. Use at least twice. And then after these exceptions. A retail establishment may make single use carryout bags made of plastic that are exempted in paragraph B available. To customers to bag products within retail establishment other than at the point of sale. Only if the retail establishment. Locates inside the retail establishment other than 20 feet of the main entrance of the retail establishment. A receptacle for collecting any single use carryout play plastic bags and ensures that single use carryout bags made of plastic that are collected by the retail establishment. A recycled or delivered to a person engaged in recycling plastics. Then you get into a section on fees. Whoops, didn't need to jump quite that far. Okay, so retail is made charge of 5 cent per bag. Fee. If they use a recycled paper bag or reusable plastic bag. And then an exception to the fees. So, I'm going to give you a little sense of what the current law is. There's no secret that some people didn't like that when it was first enacted. It was by no means unanimous. So we go back to the bill. And it appears that what this bill is trying to do. Is to go back to the prior law because when the current law, this one was enacted in 2019. Replaced this law. This bill puts that old law back and then down here. Repeals the current one. And the old law said that a retailer may use plastic bags to bag products at the point of retail sale only if the retailer. Locates a receptacle. And ensures the plastic bags are collected. Recycled or delivered to person engaged. So, um, I think in the wisdom of the legislature. This old law was not sufficient to reduce. Sufficiently the number of plastic bags that end up in. The trash or on the streets or on beaches. So forth and so on. Thus. This law. I dare say. This bill will not pass. We'll see. Thank you. I'm responding to Mandy's question, which pointed out. Oops, sorry, I was muted. I'm just responding to Mandy's question that there are actually three of those bills for appealing. And I, um, I know you'll show it at the end, but I thought that was a good opportunity to. Use the site to search. So I searched plastic bag ban on that link I sent. And. I now see those three bills. LD 39, which we were looking at LD 108. And 108 and 244. And 108 uses. You know, the language of public safety as, um. To try to improve public safety. But I just wanted to point out, thank you. Bring that up. There are a couple other bills in this regard and we do track. We do track as many of these as possible. So we will definitely be adding these to the list. This appears to me to be identical, except for the title. To the first bill we looked at. I don't see much difference. I jumped back and forth between the two. I don't see much difference. Yeah, that's an example of how the revised or didn't screen out identical bills. Let's take a look at that third one. Yeah. And I haven't read through all of these. Isabella just brings up if these are, if it is, has been effective or not. I thought there was an implementation piece. Maybe you covered it Bob of. I don't know if it's been phased in or if it is just a charge on plastic bags, but it is definitely an interesting point about. How to go about either banning or taxing. And sometimes. I know for taxes or bands. I believe I'm no legal expert, but the US constitution sometimes comes into play with the commerce clause. So some, I don't know if Bob, if you agree with that, I know sometimes people have to decide which way to go. If you're kind of taxing or banning and it has, it can have bigger implications if they're national corporations. Well, yeah. And what you will sometimes hear on that last point is that all of the state of Maine shouldn't do this. This really should be a national policy. So things are uniform from state to state. And that's what same lobbyists when Washington telling the federal government not to do it either. Now we have three identical pieces of legislation. We wouldn't have seen this before the 130th legislature but blame it on COVID but it's a schedule for hearing at the same time. The sponsors will all stand up in a single hearing, make their pitches. And the committee will, will kill two of the bills. They may kill all three of them, but, but they won't report more than one out. Great. Well, thank you. Yeah, thanks for bringing that up. And thanks Isabella for kind of pointing more towards how, how people are actually trying to address issues and are they effective is definitely something that we continue to look at for everything moving forward. I wonder if this is also as we're kind of moving towards the end, if we could look at the offshore wind ban before we start. Finishing up. Right. Did you want to talk at all about right choice voting. I not, I think that that one got the least amount of votes, but Okay. And then let me give you, let me give you 15 seconds on it. The reason this is a resolution to amend the Constitution is that right choice voting for the governor of Maine and the legislature cannot be used because the Constitution says the winner is the one who gets a plurality. That does that prohibition doesn't stand in a way of ranked choice voting for federal offices or for primary elections because those are actually party events that apply to the governor in the general election. Got to change the Constitution. All right, when do you have the bill up ahead? I think it's one, I think it's one on one right there. One on one. I did a word search on. Okay, so this, if you were to look at the sponsors down here. It's a very similar cast of characters that you will see on the plastic ban repeal. And it's interesting. Most of these legislators. Well, half of them anyway. Don't live near the oceans, but there's a couple of them that probably represent a fisherman. So anyway, here we go. We'll look at the bill itself. Fairly short bill. Let's go to the summary of proof prohibits any department or agency or political subdivision of the state like a town or city from approving or permitting offshore wind projects. And then it says that these various state departments have to submit by the end of this year. Legislation necessary to align those provisions of law under their respective jurisdictions with the prohibition and offshore wind projects. And then it says what, what, what kinds of projects are these include community based offshore wind energy products, deep water offshore. Offshore wind energy demos. And off and wind power projects. So, I'm not quite sure what their objection is. The strongest objections I think have come from. Is that the state is going to be living and in fact, the governor is going to ask the legislature for 10 year moratorium on offshore wind that's within. The three mile limit that the state regulates her proposal would not apply. To things further offshore. That. That one needs only the federal. Government to approve. That one needs only the federal government to approve. She couldn't do that anyway. She couldn't go be out beyond three miles just for that reasons. That's how far the state's reach goes. Now representative walking and he is from a coastal community. And he's a primary sponsor. So I'm sure he's hearing from his officials. Fisher folk. Bob, I'm, I'm first, I'm struck by. I don't see any. I don't see any reasoning in there. I don't see any reasoning in there. And given you would think they're in some defense and a court of law later. It would say, you know, in order to protect such and such. This is no longer allowed, but I don't see any reasoning in there. Do you. No. And it's not necessary in order to enforce a law. That this can happen and that can't. And you may find a rationale in the legislative record, but you won't find it usually in the. The law per se. You can bet that once this bill gets to a public hearing. In fact, I'm going to show you. I'm going to take the liberty of showing you. Something from the last legislative session as an example of what you can find. On a. I'm just going to pick something at random here. So all weather tires and automobiles, what the heck. So I'm going to. What happened with this is that it was given. This is not a good example because it looks like it never got a public hearing the sponsor. LTW needs leave to withdraw. I'm not sure we drew it before it ever got a public hearing. Not a good example of what I wanted to show you. Sceptic inspection of the shoreland area. We'll try that one. Okay, this is, this is good. This is the status and committee page. The committee name. When it was reported out and how, and I'll get to that in a second. Anybody. Okay. I'm going to go through that written testimony. For a bill and you can do it through a web portal. In this case, 20 people did so. Are there names. They're affiliations. You click on the names. You can. See what they have to say. Very open process. You won't, you won't get anything like this. This is not this, this easy. So, Nick Bennett, staff, scientists for N. R. C. M. Testified in support. Gives us reasons.  You can get to that question. Matt, what the basis was, what the reason was. Not so much by looking at the law, but looking at what people said to justify the law. up here. The 13 members of committees, we talked about this in early workshops, don't always agree. Often there's more than one report, and indeed there was on this bill. The majority of the 13 supported the bill. Two of them didn't like anything about it and voted against it. You can then find out what happened when the bill got to the House and Senate. Tells you when it was enacted, when it was signed to the governor, and what public law chapter number it was assigned. I don't want to keep you too late. I'm good as long as you are. Where would you like to go next? Just to, I guess, answer Bruce's question, which is definitely a legal question in my mind, but he said, what is legislative intent? Does it have any weight when rules are made, or if there are disputes in the interpretation of a law derived from the bill? I'm sorry, the last part about interpretation, I didn't miss the word. Does legislative intent have any weight when rules are made, or if there are disputes in the interpretation of a law derived from the bill? The answer is yes and yes. Legislative intent has everything to do with how the rules are drafted, but yes, that can be disagreement over the intent of both the law and the rule, and that's why we have courts to arbitrate those disputes and decide to discern what the intent of the legislature was, and the courts can look at more than just the words of the law itself. They may very well go into the legislative history and look at, it's not the one I wanted, I may have closed it, but we were looking at something a minute ago that showed a bill from the last, oh wait a minute, is this it? Yeah, the courts may very well look at this testimony. There is also something called the legislative record. Every single word spoken on the floor of the House and the Senate, at least when they're on the record, and not talking about when they're going to meet for lunch, is recorded and transcribed, and it used to be those, you could go to the law library and find those hefty tomes now they're online, but the debate on a bill is also very much fertile ground for a court to try to discern the legislative intent, and in fact some, it's not unusual for a legislator, and it may often be the chair of a committee that reported bill out, is to read a statement into that record to emphasize what the intent was, and what the meaning of the bill is. So that's the kind of stuff the courts will look at in addition to, you know, the black and white words on the page. Thank you, Bob. And I, yeah, it does strike me that as you said it's still the judge's decision, and at least when I think of, you know, other courts at the Supreme Court, they may or may not care about intent. So yeah, I guess this this was great. We got to cover all the ones we voted for, so I'm glad we could do that. I know a couple of people have jumped off, I think they probably had to leave right at the top of the hour. So I'm going to stop recording, and anybody who wants to receive the Friday reminder


Hey everyone welcome back to another undead world hero survival episode so today we're going to have a part two of my team comp suggestions okay so feel free guys to experiment with these lists and these teams they are not really sure way of winning anything but this is just a guide on based on what i've tested so far based on my own opinion and based on what i think can help you guys in terms of putting up your own teams okay so what i did was for the squads i listed down and made a couple of squads we have um to your right the right column the sky campaign and the pvp these are my old teams that i was able to discuss with you um uh last time in the last episode for those who haven't seen that episode i'm gonna put a link up there so that you can see those lineups that we discussed um the new line lineups are actually here so one two three four five six so these are going to um what are we going to be discussing so we'll start with the pvp so i have two two new pvp lineups that i was able to put together so first is going to be this lineup um this lineup uh i've tested it it's actually good um the purpose of this lineup is to have three uh three heroes that resurrect so you have in your front uh tanya you have gabriel you have helen then at the back you have elita for the buffs and you have astra there for your uh what do you call this for your resurrection as a zombie the next pvp lineup that we have here is uh pvp new too so this one i'm pretty much excited because um this team i built this and i'm using this right now in terms of pvp um sometimes a switch but now this is my main pvp which i have a new hero uh for bella um i only have one tank now i have two in the mid then two at the back so the same astra here leta then you have murry then you have bella here then in the front uh you have their um abigale as well as your tank so let's try both of these lineups in pvp take note they're only the the pvp that i mean is um for those teams are only for novice arena you can actually use them in um in the rumble the raging rumble but they are far they are more effective um yeah in the novice arena so let's take a look at okay let's change up the formation let's do this because you only have one tank technically and these two are melee let's go with this line up with with this formation and see what happens okay so again um as expected they lose their you know their hp as fast as possible but they will be resurrected while being resurrected obviously um your enemies are going to lose a couple so hopefully we could turn the tide of this uh of this battle you have their resurrection again for helen and still we are in battle so still neck and neck again these heroes um especially my front line like helen and gabriel don't have that much good gear but again if you prefer a more heavy on the melee side of things in terms of pvp um it's actually a good team i tried them out before the next one while we have um stale tickets for pvp um let's do the other pvp that we'll be testing is going to be this one so this involves already um bell uh sorry bella let's try this out at least you can see how you know how it goes this one is a more you know for me it favors my play because i have range i have a i have a a tank and also i have melee and bella is actually perfect for pvp as you can see this team is more uh effective and balanced and i built in in fairness this team i have built very well already the next is um we're gonna do look at is the train so train in terms of zombie express i have a new lineup for zombie express that i am actually using right now so for zombie express um is this the lineup so train you okay so this is the lineup so i have you have there abigail the front then you have um you have a lot of a we damage dealers at the back so more on buffs because what you aim to do in the train is to do a lot of damage as quick as possible to gain more points um earlier stages of the train or zombie express is going to be more on the defensive side but as you progress you're gonna have to um advance your teams on the offensive side so abigail if you have another aoe um hero you can actually replace abigail uh with another aoe hero that you have um i'm also thinking of adding hooky here uh to this team so hopefully it would you know give me more aoe damage so so far i'm up to i'm gonna end hopefully at two million at the one minute mark as you can see there so i'm on my way to seven i actually play tested this a couple of times so it usually ends up at seven million um but my goal right now is to probably go beyond seven million to the next level i think cookie would bring me there especially with her buffs with her aoe damage as well but i actually have her still at the most basic level at epic so 20 seconds left to go we are five million and we are nearing 7.5 which is my high score hopefully i can actually beat that score but this is actually going to be very close yes i'm at 7.7 for this run so a new high score so as you can see um for for your zombie express i keep calling it the train zombie express is going to be more on aoe damage um damage multipliers damage buffs as you can see there that is the squad that i actually was using so we're done with two new pvp lineups you have the um the train again then the rest is going to be campaign and skyscraper lineup so we have three um please bear with me because the current skyscraper lineups that i have here here is actually very tough even my my own current team i cannot even you know finish this yet so i am at level or at floor 174 the squad that i'm using usually is this one so this has been you know has taken me to this level already this has is or is or was in my previous video the new ones that i've been experimenting on involves this one let's go more with let's start with this one so this is actually another lineup that i was experimenting so what the goal for this one is just to pass the first wave so as you can see this lineup is tailored fit war for sugar reapers with a splash of splash of cookie as you can see cookie was pulled in that is a problem with this stage they keep on pulling in healers towards them that is why healers usually don't survive this sorry not healers um but for the lowest hp that you have they keep on pulling it towards them but again this lineup survived the first wave which is actually promising and this can be tweaked further more as you can see cookie here is at epic i just need to give her more or get more copies of her let's end this let's see the others i have two other lineups to show you for skyscraper in campaign so the next lineup that we have here that i'm actually experimenting on is this one so this one is more on the concept of having more colligium here which is connor is here and the rest is going to be your blue banshees so again this is going to be a lineup which is there's only one tank so let's see if this lineup can survive the first round again even connor i haven't given good gear yet um that is why i don't expect him to last that long as you can see there um again this lineup i it's a bit more experimental at this point but this lineup once you have you know have tweaked it to a point where in the gear is good this lineup is going to be good as well for skyscraper and the last lineup that we're going to be discussing is this is basically very experimental but i think this is one of the you know i think best experiments that i've i've had i'm so far i've been experimenting with heroes that haven't been used so that is why lee is here there are a lot of you know the other people who don't use lee um i've i've seen one comment that lee should be topped here so i tried using her and this lineup actually is an example that she is good but again with the lowest hp i think of the bunch she's going to be pulled again for the stage so let's see let's see what happens so three at the back one mid and one the front so again for this lineup this is going to have a lot of damage as you can see i dwindled it dwindled it down to down to 11 so for the you know the the smallest hp it was um rock was pulled in but the rest can actually survive wave one i hope so so as you can see front line you have abigale you have tanya they survived they haven't even died or been resurrected yet but again we cleared wave one except for rock being dead so that's it another experimental team that you can actually build on hopefully um you try out some of the lineups that we've discussed today and hopefully guys um i actually appreciate it if you give me feedback if you if you use some of the lineups what do you think are they good um do you have any suggestions can put them down in the comment section below also for those who um have gone this far please do consider subscribing if you haven't subscribed yet also don't forget to click that bell icon and put a like to this video thank you guys stay safe take care this is the warden and i'm out of here