This post will be the first of three parts, culminating in a proof of the fundamental theorem of Galois Theory:

1) Finite Separable Algebras

2) Galois Connections

3) The Fundamental Theorem of Galois Theory

These posts are designed to aid in the third part, so, for the reader who is looking for the easiest route to a proof of the fundamental theorem, I have put a star (*) next to the essential parts of this post.

The (elementary) fundamental theorem of Galois theory asserts a correspondence between sub-extensions of certain field extensions and subgroups of . Those field extensions for which the theorem holds are appropriately named the Galois field extensions. It is this theorem that motivates me to make this post, but in particular I am motivated because I want to show that this theorem is simpler and altogether more manifest than many expositions would lead you to believe. In point of fact, there are only two challenging theorems at play here, depending on where certain lines are to be drawn. One of these is the fundamental theorem of Galois theory, and another is some equivalent of Artin’s linear independence of characters. In spite of this, some expositions of this fundamental theorem in its most elementary form make it appear to be more tricky than it is. For instance, some authors use primitive element theorem where it isn’t necessary.

In this first post, though, we talk about finite separable algebras. We’ll define the concept of a finite -separable -algebra. Every finite -separable -algebra will be finite -separable, and we’ll call a -algebra separable if it is -separable.

Throughout, we set as a field and as a finite field extension of . For -algebras and , we write for the set of -algebra maps from to . We write for the algebraic closure of . Throughout, we set . For a group and -sets and , we write for the set of -set maps from to .

We write for the group algebra over . For a -action , write for the -module consisting of formal sums where all but finitely many of are zero. The structure map is determined by specifying that . Write for the -module consisting of formal sums viewed as functions . The structure map is determined by sending to the function sending to .

Our first result plays the role of Artin’s linear independence of characters in his rendition of Galois theory. The interested reader might examine the proof of (i) below to look for a commonality with Artin’s proof.

**Theorem: **

(i) For each -algebra , there is a canonical map of -modules, which sends to the map sending to . is an injection.

(ii)* For each -set , there is a canonical map of -modules, which sends to the map sending to . is an injection.

(iii)* For each -algebra , there is a canonical map of -modules, which sends to . is a surjection.

(iv) For each -set , there is a canonical map of -modules, which sends to . is a surjection.

**Proof: **

(i) Take in the kernel of , such that the amount of nonzero is the least possible. Take , and take such that for some with . Then for each . And , so is contained in the kernel of . Yet this element is nonzero, as and were chosen so that for some , and . So we have a nonzero element of the kernel of with strictly fewer nonzero summands, a contradiction.

(ii) Take in the kernel of , such that the amount of nonzero is the least possible. Choose and such that and . Then for each . And , so is contained in the kernel of . Yet this element is nonzero, as and were chosen so that for some , and . So we have a nonzero element of the kernel of with strictly fewer nonzero summands, a contradiction.

(iii) For each , there is a map sending to with kernel . are coprime ideals as they are distinct maximal ideals, so they produce a surjective canonical map by the Chinese remainder theorem.

(iv) For each , there is a map sending to with kernel . are coprime ideals as they are distinct maximal ideals, so they produce a surjective canonical map by the Chinese remainder theorem.

**Corollary: ** For each finite dimensional -algebra , we have . Then if and only if is an isomorphism.

**Proof: ** is a surjection, so that is an isomorphism if and only if . But and , so is an isomorphism if and only if .

This sets up the stage for our primary definition, and topic of the hour: finite separable -algebras. If always, then when do we have ? We start by putting a name on this condition:

**Definition:*** We say a finite dimensional algebra is -separable if . We say that is separable if it is -separable (we get the same number if we use the separable closure, if you know what that is).

I refer to as the ‘geometric dimension’ of . Then being separable says that the geometric dimension of is the dimension of .

**Proposition: ** Let be a field and a field extension of . A -algebra is -separable if and only if is a free -module of dimension .

**Proof: ** first suppose that is a free -algebra of dimension . Then there is an isomorphism . . (That is left as an exercise). So that is separable. If is separable, then is an isomorphism, so that is a free -algebra of dimension .

Lastly, I wanted to show some stability results for finite -separable -algebras

**Proposition: ** Let and be finite -algebras. We have the following stability results for -separability:

(i) If and are finite -separable -algebras, then is finite -separable.

(ii) If and are finite -separable -algebras, then is finite -separable.

(iii)* If is finite -separable and there exists an injective map , then is finite -separable.

(iv) If is finite -separable and there exists a quotient map , then is finite -separable.

(v) If is generated by finitely many finite -separable subalgebras , then is -separable.

(vi) and are finite -separable.

**Proof: **

(i) Suppose and are finite -separable -algebras. The functor preserves coproducts. In fact it preserves all colimits, as it is left adjoint (proof of this, in whatever prefered generality, is left to the reader). The upshot is that the canonical algebra map is an isomorphism. Take isomorphisms and . We then have

Thus is a free -algebra of dimension , so that is separable.

(ii) Suppose and are finite -separable -algebras. We may make a similar argument to before to show that the canonical map is an isomorphism. Take isomorphisms and . We then have

Thus is a free -algebra of dimension , so that is separable.

(iii)* Suppose is finite -separable and take an injective map of -algebras. The following diagram commutes:

is injective since is a flat -module; all -modules are flat since is a field. From this it follows that the map is injective, so that is separable by the theorem above.

(iv) Suppose is finite -separable and take a surjective map . To show that is -separable, we show that is a free -algebra of dimension . The composition is a surjection. Take to be the -dimension of the kernel of this map, so that . We have

so , so .

(v) It suffices to show that, if is a -algebra generated by -separable -subalgebras and , then is -separable. In this case, there is a surjective canonical map sending to . is -separable by (i), and by (iv) this implies that is separable.

(vi) This is clear.