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.