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Question on normal field ext.

  1. Dec 12, 2007 #1
    If g(x)[tex]\in[/tex] K[x] and 1< deg(g)=n.
    Given that G/K is a normal field ext., if g(x)=g1(x)*....*gk(x)[tex]\in[/tex] G[x],
    then deg(g1)=....=deg(gk)

    My attempt:
    I let G = K adjoins the coefficients of gi's.
    Let [tex]\alpha[/tex] be a root of g.
    Notice that K [tex]\subseteq[/tex] G [tex]\subseteq[/tex] K( [tex]\alpha[/tex]) = G( [tex]\alpha[/tex])

    We can express g(x) = irr( [tex]\alpha[/tex], G) *p(x) for some p(x) [tex]\in[/tex] K[x].
    if i let g1 be irr( [tex]\alpha[/tex], G),
    then deg(g1) = deg( irr( [tex]\alpha[/tex], G)) = [G([tex]\alpha[/tex]):G]=[K([tex]\alpha[/tex]):G]

    Next, we do the same thing again with another root, say [tex]\beta[/tex].
    g(x) = irr([tex]\beta[/tex], G) *q(x) for some q(x) [tex]\in[/tex] K[x]
    if i let g2 be irr( [tex]\beta[/tex], G),
    then deg(g2) = deg( irr( [tex]\beta[/tex], G)) = [G([tex]\beta[/tex]):G]=[K([tex]\beta[/tex]):G]

    if we proceed in the same way, deg(gi) can be found to be equal to [K([tex]\theta[/tex]):G] for some root [tex]\theta[/tex] of g.

    Next, we observe that since K([tex]\alpha[/tex]) and K([tex]\beta[/tex]) are normal extnesion of K, they split into linear factors for g(x). so K([tex]\alpha[/tex]) = K([tex]\beta[/tex]) (and this in fact implies to K adjoining other roots of g)

    so deg(g1)=......=deg(gk)

    please tell me if there's anything wrong with it.. this question is a bit too advanced for me :confused:
  2. jcsd
  3. Dec 12, 2007 #2


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    Homework Helper

    I suppose we're assuming that g(x) is irreducible over K. I skimmed through your solution, and I think you have the right ideas, but your write-up is very hard to read - and some things don't make sense, e.g. what do you mean when you say "K(a) splits into linear factors for g(x)"? If I'm not mistaken, this problem is from Chapter V of Hungerford's Algebra, correct? It was one of my favorite problems!

    Here are some tips that will help make your post more readable: When you want to use inline TeX (i.e. during sentences), use [itex] instead [tex]. This will make everything align nicely. Also, try to state clearly what you're trying to show at each step.

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