TITLE: The fundamental theorem of Galois theory
QUESTION [12 upvotes]: Who proved the modern form of the fundamental theorem of Galois theory?. Was it in the original Galois' manuscript?

REPLY [9 votes]: 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.