# The fundamental theorem of Galois theory

Who proved the modern form of the fundamental theorem of Galois theory?. Was it in the original Galois' manuscript?

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According to en.wikipedia.org/wiki/Field_theory_(mathematics), the definition of an abstract field wasn't written down until 1893 (by Weber). So I doubt Galois had the tools to think in such generality. –  Qiaochu Yuan Feb 10 '12 at 6:26
The question is very close to mathoverflow.net/questions/26811/galois-theory-timeline –  Alain Valette Feb 10 '12 at 8:08
Duplicate? See mathoverflow.net/questions/26557/… –  Alain Valette Feb 10 '12 at 13:15
@Alain: It is close but not a duplicate. Although the contribution of Artin is significant, the fundamental theorem could be proved before. In fact, as Steven Landsburg answers below, it follows from Galois own results easily. –  Mark Sapir Feb 10 '12 at 13:40
Many thanks to everybody for your answers! I think that Steven's answer is what I was looking for, so I accept it. –  Mark Sapir Feb 10 '12 at 13:55

Galois's Proposition I (as translated by Edwards) is:

Let the equation be given whose $m$ roots are $a,b,c,\ldots$. There will always be a group of permuations of the letters $a,b,c,\ldots$ which will have the following property: 1) that each function invariant under the substitutions of this group will be known rationally; 2) conversely, that every function of these roots which can be determined rationally will be invariant under these substitutions.

As Edwards observes, it takes a lot of work to decipher the exact meanings of "substitution" and "invariant" here, but once you've done that, this can be translated into modern language as:

If an element of the splitting field of $K(a,b,c,\ldots)$ is left fixed by all the automorphisms of the Galois group then it is in $K$.

The fundamental theorem of Galois theory (i.e. the Galois correspondence) follows easily, though Edwards doesn't say who first stated it.

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For Galois' original text (published in Journal de Liouville), see p. 421 (scroll to the 2/3 of the document): portail.mathdoc.fr/JMPA/PDF/JMPA_1846_1_11_A47_0.pdf There is an interesting footnote (is it also in Edwards?) where Galois tries to explain what he means by "invariant" and "known rationally". It certainly agree with the comment: "It takes a lot of work to decipher..." –  Alain Valette Feb 10 '12 at 14:36
@Alain: Your point is of course valid. The problem is that Galois did not have the modern terminology (also time!), he needed to invent many things from scratch. Similarly, if you read what Euclid actually wrote and compare with what we think he wrote, you will find that Euclid's texts are very hard to decipher also. There are much more modern (the second half of the 20th centry) examples which you certainly know. –  Mark Sapir Feb 10 '12 at 15:54
@Mark: This is why (with due respect) I am not quite satisfied with Steven's answer: someone had to do the deciphering work, and I wonder who it was (being aware that it might not be a single person). A Google search points to contributions by Noether, van der Waerden, and Artin. –  Alain Valette Feb 10 '12 at 19:53
@Alain: As I understand, it took about 100 years after Galois to translate what he did into the modern terminology and many people contributed to it. Qiaochu Yuan mentioned Weber. Also E. Artin, Van der Waerden, and many others. I find this text very interesting: gap-system.org/~history/Projects/Brunk/Chapters/Ch3.html –  Mark Sapir Feb 10 '12 at 21:58

Surely the book

Edwards, Harold M. Galois theory. Graduate Texts in Mathematics, 101. Springer-Verlag, New York, 1984. xiii+152 pp. ISBN: 0-387-90980-X MR0743418 (87i:12002) (link to amazon)

would be useful. This book about Galois theory is made from the historical point of view. In particular, it contains an English translation of Galois' memoir.

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The modern form can be found in Galois Theory, by Emil Artin and Arthur N. Milgram page 46, published in 1944. I'm not expert in math history, but once, I heard Artin was the first person to wrote the modern account of Galois theory.

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-Yes, but there were many proofs earlier(and modern) for the fundamental theorem of Galois theory before Emil's. Emil's proof is simpler than the proof the usual ones we find in every mongraph –  Ongaro Nyang' Feb 10 '12 at 11:29

I am not particularly interested in mathematical history, but Peter Newmann is: http://www.ems-ph.org/books/book.php?proj_nr=137.

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