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A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective (which based on past teaching I don't entirely share) that the periods worth considering are pre-Artin, classic Artin treatment, and post-Artin. To make the point explicitly, that is to do with the influence of Artin's Galois Theory Notre Dame notes, copyright dates 1940 and 1942.

My issues with this periodisation are primarily to do with a wish to have a proper view of innovation, starting with Galois (admitting pre-history evident in Gauss and Abel, solution of the quartic, group theory and other contributions in Lagrange). There is something like this:


*Liouville writes up the theory

*French school of group theory and treatment by Camille Jordan

*Riemann surface theory in general, and isogenies of elliptic curves in particular, develop in parallel

*Presumably Hurwitz knew how to connect the dots

*Algebraic number theory uses abelian extensions and Kummer theory extensively

*Hilbert lays conjectural foundations for class field theory, post-Kronecker Jugendtraum and complex multiplication theory, using a version of Galois theory that seems to be much influenced by Hurwitz/Riemann surfaces

*Steinitz, abstract theory of fields, idea of separable extensions clarified

*New expositions from Emmy Noether and Artin in the 1920s (are these documented, though?), against the background of completing proofs of class field theory, and Artin L-functions

*The Inverse Problem for Galois groups is stated and leads to work on invariant theory

*1930s: Galois theory for infinite extensions is enunciated

*C.1940: Tensor products of fields.

This takes us just about to 1940. I think it is a trap to assume Artin in 1940 was lecturing on Galois theory in the precise terms he would have used in the 1920s.

I'd be grateful for help making this tentative timeline more solid. Further interesting things did happen after 1942, but that seems enough for one question.

[Edit:The older question was What was Galois theory like before Emil Artin? - treat my remarks there as tentative.]

Edit: Dedekind's contribution should have been on the list. See about what Dedekind did in his Vorlesungen. That article credits Artin with the formulation of the Fundamental Theorem in abstract terms, while crediting Dedekind with the theory for subfields of the complex numbers. In that context it becomes clearer, I think, why "innovation" goes to Dedekind on the lattice-theoretic way of thinking about Galois theory, while Artin might reasonably have thought he was doing exposition (bringing those ideas explicitly into the post-Steinitz era).

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I prefer to distinguish between pre-Steinitz (Galois theory as a theory of solving equations) and post-Steinitz (Galois theory as a theory of field extensions). As for Emmy Noether's input, there are a couple of remarks on the theorem of primitive elements in her correspondence with <a href="">Hasse</…; – Franz Lemmermeyer Jun 2 '10 at 13:25
Who proved the normal basis theorem? – Charles Matthews Jun 2 '10 at 16:59
Charles: See… – KConrad Aug 14 '10 at 14:19
Apparently there are speculations about "forgotten" concepts of Galois, maybe in relation to Galois theory:… – Thomas Riepe Dec 14 '11 at 8:48

2 Answers 2

EDIT. Here is the part of the answer that has been rewritten:

We give below a short proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below).

The derivation of the FTGT from (a) and (b) takes about four lines, but I haven't been able to find these four lines in the literature, and all the proofs of the FTGT I have seen so far are much more complicated. So, if you find either a mistake in these four lines, or a trace of them the literature, please let me know.

The argument is essentially taken from Chapter II (link) of Emil Artin's Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin's proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin's argument) was written with Keith Conrad's help.

Theorem. Let $E/F$ be an extension of fields, let $a_1,\dots,a_n$ be distinct generators of $E/F$ such that the product of the $X-a_i$ is in $F[X]$. Then

  • the group $G$ of automorphisms of $E/F$ is finite,

  • there is a bijective correspondence between the sub-extensions $S/F$ of $E/F$ and the subgroups $H$ of $G$, and we have $$ S\leftrightarrow H\iff H=\text{Aut}(E/S)\iff S=E^H $$ $$ \implies[E:S]=|H|, $$ where $E^H$ is the fixed subfield of $H$, where $[E:S]$ is the degree (that is the dimension) of $E$ over $S$, and where $|H|$ is the order of $H$.


We claim:

(a) If $S/F$ is a sub-extension of $E/F$, then $[E:S]=|\text{Aut}(E/S)|$.

(b) If $H$ is a subgroup of $G$, then $|H|=[E:E^H]$.

Proof that (a) and (b) imply the theorem. Let $S/F$ be a sub-extension of $E/F$ and put $H:=\text{Aut}(E/S)$. Then we have trivially $S\subset E^H$, and (a) and (b) imply $$ [E:S]=[E:E^H]. $$ Conversely let $H$ be a subgroup of $G$ and set $\overline H:=\text{Aut}(E/E^H)$. Then we have trivially $H\subset\overline H$, and (a) and (b) imply $|H|=|\overline H|$.

Proof of (a). Let $1\le i\le n$. Put $K:=S(a_1,\dots,a_{i-1})$ and $L:=K(a_i)$. It suffices to check that any $F$-embedding $\phi$ of $K$ in $E$ has exactly $[L:K]$ extensions to an $F$-embedding $\Phi$ of $L$ in $E$; or, equivalently, that the polynomial $p\in\phi(K)[X]$ which is the image under $\phi$ of the minimal polynomial of $a_i$ over $K$ has $[L:K]$ distinct roots in $E$. But this is clear since $p$ divides the product of the $X-a_j$.

Proof of (b). In view of (a) it is enough to check $|H|\ge[E:E^H]$. Let $k$ be an integer larger than $|H|$, and pick a $$ b=(b_1,\dots,b_k)\in E^k. $$ We must show that the $b_i$ are linearly dependent over $E^H$, or equivalently that $b^\perp\cap(E^H)^k$ is nonzero, where $\bullet^\perp$ denotes the vectors orthogonal to $\bullet$ in $E^k$ with respect to the dot product on $E^k$. Any element of $b^\perp\cap (E^H)^k$ is necessarily orthogonal to $hb$ for any $h\in H$, so $$ b^\perp\cap(E^H)^k=(Hb)^\perp\cap(E^H)^k, $$ where $Hb$ is the $H$-orbit of $b$. We will show $(Hb)^\perp\cap(E^H)^k$ is nonzero. Since the span of $Hb$ in $E^k$ has $E$-dimension at most $|H| < k$, $(Hb)^\perp$ is nonzero. Choose a nonzero vector $x$ in $(Hb)^\perp$ such that $x_i=0$ for the largest number of $i$ as possible among all nonzero vectors in $(Hb)^\perp$. Some coordinate $x_j$ is nonzero in $E$, so by scaling we can assume $x_j=1$ for some $j$. Since the subspace $(Hb)^\perp$ in $E^k$ is stable under the action of $H$, for any $h$ in $H$ we have $hx\in(Hb)^\perp$, so $hx-x\in(Hb)^\perp$. Since $x_j=1$, the $j$-th coordinate of $hx-x$ is $0$, so $hx-x=0$ by the choice of $x$. Since this holds for all $h$ in $H$, $x$ is in $(E^H)^k$.

[A] Emil Artin, Galois Theory, Lectures Delivered at the University of Notre Dame, Chapter II, available here.

PDF version:

Here is the part of the answer that has not been rewritten:

Although I'm very interested in the history of Galois Theory, I know almost nothing about it. Here are a few things I believe. Thank you for correcting me if I'm wrong. My main source is

Artin was the first mathematician to formulate Galois Theory in terms of a lattice anti-isomorphism.

The first publication of this formulation was van der Waerden's "Moderne Algebra", in 1930.

The first publications of this formulation by Artin himself were "Foundations of Galois Theory" (1938) and "Galois Theory" (1942).

Artin himself doesn't seem to have ever explicitly claimed this discovery.

Assuming all this is true, my (probably naive) question is:

Why does somebody who makes such a revolutionary discovery wait so many years before publishing it?

I also hope this is not completely unrelated to the question.

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The .pdf link at the end is broken. – KConrad Oct 21 '11 at 14:52
Dear @KConrad: I've just noticed your comment. Thank you very much! (The link should work now.) – Pierre-Yves Gaillard Dec 7 '11 at 14:59
I like your write-up, but its not related at all to the question (which is about history). – Martin Brandenburg Dec 14 '11 at 9:49
Dear @Martin: Thanks! I agree. In the previous versions of the answer, what is now Part 1 was Part 2. Old Part 2 was much shorter and added as a loosely related complement. --- I'm kind of obsessed by the question of knowing if my argument is correct. If it is, it's seems hard to believe that it had not been written somewhere before. It's true that I'm taking advantage of the post to ask this question. – Pierre-Yves Gaillard Dec 14 '11 at 10:31
A related answer is at…. – KConrad Dec 14 '11 at 14:48

There is a new book, L Martini and L Toti Rigatelli, The Algebraic Mind: Galois Theory in the 19th Century. I haven't seen the book, just the Birkhauser catalog announcing it for Summer 2010. "The aim of this book is to discuss the development of Galois theory from 1830 to the turn of the twentieth century...."

I was unable to find any trace of the book on the birkhauser website.

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There do seem to be books. One of my "issues" is this: do the historians of Galois theory "read back" what the scope of the theory is supposed to be, from the way it looked in say 1950? The explicit cases of elliptic curve isogenies of multiplication by n, and maybe coverings of modular curves, were well known to Weber (another answer). We know about "Galois theory of coverings" now: how did it look then? – Charles Matthews Jun 3 '10 at 7:27

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