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Consider the names of basic algebraic structures: 'group', 'ring', 'space', 'field', 'Körper', even the name 'structure' itself - all of them time-honoured terms, deeply rooted in our history and culture.

But what has an algebraic field to do with an acre? What has an algebraic group to do with a group of people?

Even when it's known who coined these names (of algebraic structures), it's not obvious why they were choosen and what the connection is between the named structures and what was named originally (or later on). Only those who coined the names could tell.

Are there etymological studies concerning these names - 'group', 'ring', 'space', 'field',... - which elucidate this connection?

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Even worse is the word "flat" as applied in commutative algebra, which seems like it ought to be justified geometrically but is not--or at least, not easily (there's an interesting question about this). Another note is that the German and French words for "field[mathematics]" actually mean "body." The German term came first, coined by Dedekind, who used it for subfields of $\mathbb{C}$; I'd be curious to know how this ended up being called a "field" in English. –  Charles Staats Aug 11 '10 at 22:56
Your curiousness is mine. (I should have given the hint that 'Körper' means 'body'/'corpse'.) –  Hans Stricker Aug 11 '10 at 23:02
Conversely, Germans used "Feld" for "field" (as in "Galoisfeld", something which is nowadays widely considered an anglicism but actually appears e. g. in Witt's works). –  darij grinberg Aug 11 '10 at 23:35
What about "domain"? Was there a time when the phrase "integral domain" was not redundant? –  Tom Goodwillie Aug 12 '10 at 11:40
And what about all the agricultural terminology in algebra and algebraic geometry: kernel, fiber, bundle, stalk, germ, sheaf, roots, etc.? –  Alan Guo Jan 9 '11 at 8:17

5 Answers 5

Ring came from Zahlring, which was Hilbert's term for what we would essentially call a ring of algebraic integers. Dedekind earlier used the term ordnung (= order, taken from the Linnean classification terminology like class and genus). For more on this, see the comments to the question Why is "h" the notation for class numbers?.

Fields in the algebraic sense used to be called bodies (thus closer to French and German). [Edit: In 1900, Pierpoint's "Galois' Theory of Algebraic Equations, Part II", in the second volume of Annals of Mathematics, uses "body" for field and "inferior body" for subfield, introduced on page 25. In 1910, Legh Reid's "The elements of the theory of algebraic numbers" uses the term "realm" for field, or more specifically for number field. Reid's text can be found on Google books, and on p. vi of the preface he writes that "realm" is synonymous with Körper, corpus, campus, body, domain, and field. In 1934, Heilbronn and Linfoot wrote a paper "On the imaginary quadratic corpora of class-number one", so corpus was still in use in the early 1930s.]

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+1 The explanation given under the link is way more convincing than what I was told (ring coming from cycling around in Z/nZ) –  Peter Arndt Aug 11 '10 at 23:36
Peter, did you hear it on April 1, by any chance? Even then... –  Victor Protsak Aug 12 '10 at 5:08
There are people in Italy who still use the word "corpo" to mean field, which is now ordinarily called "campo" (which literally means field). –  babubba Aug 12 '10 at 7:37
Actually Italians have it both ways: "corpo" is usually a skew-field (like in "il corpo dei quaternioni") while "campo" (i.e. "field") is a "corpo commutativo". –  Andrea Mori Oct 9 '11 at 17:20

For the origin of all names, see: http://jeff560.tripod.com/mathword.html

For the origin of all symbols, see: http://jeff560.tripod.com/mathsym.html

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Tripod! I thought they were extinct! –  Makhalan Duff Jan 9 '11 at 21:25

I'm not sure that I'm historically accurate, but that is how I always thought about algebraic nomenclature.

1) Group actually comes from group of substitutions. I guess that Galois could have introduced any other word, like "set" of substitutions or "flock" of transformations. Set theory was not yet established, so I guess a collection of functions could be called 'group', 'set' and so on according to the taste.

2) For field, I guess it comes from the meaning of field as "sphere", "subject", "area". It makes sense that such a word could come in talking about "solving an equation in the real field" rather than "solving an equation in the complex field". Then the concept of an abstract field could have followed.

3) Ring comes from "Zahlring", ring of numbers. This, as far as I know, is a terminology due to Dedekind. He was actually working with number rings, of the form $\mathbb{Z}[\alpha]$, where $\alpha$ is integral over $\mathbb{Z}$. So for some $n$, $\alpha^n$ can be expressed in terms of lower powers of $\alpha$; in some sense the components of the basis of $\mathbb{Z}[\alpha]$ over $\mathbb{Z}$ cycle, although this is accurate only when $\alpha$ is a root of unity. Hence the name ring of numbers.

4) Ideal is easy. When Dedekind realized that in a ring like $\mathbb{Z}[\sqrt{-5}]$ unique factorization does not hold, he searched for a substitute. He then realized could restore unique factorization allowing something more general than elements, the ideals. These are now called this way since he thought of them as "ideal elements" of the ring. useful to restore unique factorization. It is a fortunate coincidence that indeed for the rings he was working with (which are now called Dedekind rings), unique factorization for ideals actually holds.

5) Idéle has the same origin, being the contraction of the French "idéal élement", although the wording is inverted with respect to French use.

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In (3), alpha is integral (over Z), not algebraic. –  KConrad Aug 12 '10 at 1:53
Dedekind used the word "ideal" because he was formalizing Kummer's "ideal numbers". I believe (but don't recall the source) that Kummer was following the usage of "ideal" in geometry, where one had "ideal elements" such as complex points or points at infinity. –  John Stillwell Aug 12 '10 at 4:23
Regarding (1), I believe that "group" comes from "group of substitutions" (an English translation of the original French, I guess), where "substitutions" means the same thing as "permutations", and the objects that were being permuted were the roots of a polynomial. –  Emerton Aug 12 '10 at 4:37
Sorry, I was a bit asleep when I wrote this. I now have edited the post a bit. –  Andrea Ferretti Aug 12 '10 at 10:16
Andrea, in (4) it was not a fortunate coincidence that his rings have unique factorization? It wasn't something Dedekind stumbled onto but this was in fact the whole point. Dedekind was searching for the right context in which to make sense of Kummer's work on "ideal numbers", and, for instance, not all rings of the form Z[a] would be suitable. (Dedekind already found other problems with rings like Z[a].) The idea of considering at one stroke all the alg. integers in a number field removed the problems since they have unique. factn. of ideals. –  KConrad Aug 13 '10 at 4:02

I know that Stillwell's book "Mathematics and its History" claims that Galois introduced the word "group", though doesn't explain why he chose it.

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