The fundamental theorem of Galois theory - MathOverflow

The fundamental theorem of Galois theory

2012-02-10T05:02:14Z

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

Answer by Leandro Vendramin for The fundamental theorem of Galois theory

2012-02-10T06:27:00Z

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.

Answer by Ehsan M. Kermani for The fundamental theorem of Galois theory

2012-02-10T07:43:18Z

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.

Answer by Yiftach Barnea for The fundamental theorem of Galois theory

2012-02-10T08:59:35Z

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

Answer by Steven Landsburg for The fundamental theorem of Galois theory

2012-02-10T13:06:23Z

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.