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.