This post is the third of three parts, culminating in a proof of the (classical) fundamental theorem of Galois Theory:
1) Finite Separable Algebras
2) Galois Connections
3) The Fundamental Theorem of Galois Theory
We use the same notation as we used in (1) and (2), so read those if you don’t know any of the terms here. For the reader who is looking for the minimal route to a proof of the fundamental theorem, I have put a star (*) next to the essential parts of this post.
Definition:* Let be a finite field extension. We say is Galois if is a -separable -algebra.
Lemma: Let be a finite field extension, and let be the algebraic closure of . is Galois if and only if is separable and the canonical map is surjective.
Proof: If the canonical map above is surjective, then , so . Hence is -separable.
Conversely, suppose is -separable. Then , so , so that is separable. Since , the map is surjective, as desired.
The Fundamental theorem of Galois theory asserts that there is a Galois correspondence between intermediate fields of a Galois field extension and subgroups of . Going along with more recent approaches to the theorem, we choose here to write in terms of group actions instead of subgroups. Here we show that there is a Galois correspondence between intermediate fields of a finite Galois field extension and quotient -sets of . This exercise shows that the two approaches are equivalent:
Exercise: there is a preorder isomorphism between subgroups of a group and quotient -sets of . The correspondence sends a subgroup of to the quotient -set and a quotient -set with quotient map to , where is the identity element in .
Setup: * Let be the preorder of subobjects of in the category of -algebras. Let be the preorder of quotient objects of in the category of -sets.
For a -algebra , the set of -algebra maps from to has the structure of a -set. acts on by sending to . Define a preorder map sending a -algebra with monomorphism to the -set with epimorphism sending to .
For a -set , the set of -set maps from to has the structure of a -algebra. For two -sets and , we set to send to and to send to . For an element and , we set to send to . Define a preorder map sending a -set with epimorphism to the -algebra with monomorphism sending to .
and form a Galois connection. Indeed, taking a subobject of and a quotient object of , any map of -algebras gives a map of -sets sending to the map of -algebras sending to . Conversely, any map of -sets gives a map of -algebras sending to the map of -sets sending to .
At last, we have the fundamental theorem of Galois theory:
Theorem: (The Fundamental Theorem of Galois Theory) Let be a finite Galois field extension. Put . There is a preorder isomorphism between and given by and .
Proof: Take a -algebra . Put . Then by the main proposition in my post on Galois connections. Hence . is -separable since it is Galois (we showed this above). and are both subobjects of , and so they are both -separable, by the stability results in my post on finite separable algebras. So and . It follows that . Now the inclusion (see my post on Galois connections) is an injection of -algebras of the same finite dimension, and so must be an isomorphism.
Conversely, take a -set . Put , and . The quotient map gives an inclusion map . This inclusion tells us that is -separable, one of the stability properties in my post on finite separable algebras. So . Also, by part (ii) of the main theorem in my post on finite separable algebras. Now the canonical map is a surjection and , so that it must be an isomorphism of -sets.