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Let be a field, and let
be a finite extension of that field; in this post we will denote such a relationship by
. We say that
is a Galois extension of
if the cardinality of the automorphism group
of
fixing
is as large as it can be, namely the degree
of the extension. In that case, we call
the Galois group of
over
and denote it also by
. The fundamental theorem of Galois theory then gives a one-to-one correspondence (also known as the Galois correspondence) between the intermediate extensions between
and
and the subgroups of
:
Theorem 1 (Fundamental theorem of Galois theory) Let
be a Galois extension of
.
- (i) If
is an intermediate field betwen
and
, then
is a Galois extension of
, and
is a subgroup of
.
- (ii) Conversely, if
is a subgroup of
, then there is a unique intermediate field
such that
; namely
is the set of elements of
that are fixed by
.
- (iii) If
and
, then
if and only if
is a subgroup of
.
- (iv) If
is an intermediate field between
and
, then
is a Galois extension of
if and only if
is a normal subgroup of
. In that case,
is isomorphic to the quotient group
.
Example 2 Let
, and let
be the degree
Galois extension formed by adjoining a primitive
root of unity (that is to say,
is the cyclotomic field of order
). Then
is isomorphic to the multiplicative cyclic group
(the invertible elements of the ring
). Amongst the intermediate fields, one has the cyclotomic fields of the form
where
divides
; they are also Galois extensions, with
isomorphic to
and
isomorphic to the elements
of
such that
modulo
. (There can also be other intermediate fields, corresponding to other subgroups of
.)
Example 3 Let
be the field of rational functions of one indeterminate
with complex coefficients, and let
be the field formed by adjoining an
root
to
, thus
. Then
is a degree
Galois extension of
with Galois group isomorphic to
(with an element
corresponding to the field automorphism of
that sends
to
). The intermediate fields are of the form
where
divides
; they are also Galois extensions, with
isomorphic to
and
isomorphic to the multiples of
in
.
There is an analogous Galois correspondence in the covering theory of manifolds. For simplicity we restrict attention to finite covers. If is a connected manifold and
is a finite covering map of
by another connected manifold
, we denote this relationship by
. (Later on we will change our function notations slightly and write
in place of the more traditional
, and similarly for the deck transformations
below; more on this below the fold.) If
, we can define
to be the group of deck transformations: continuous maps
which preserve the fibres of
. We say that this covering map is a Galois cover if the cardinality of the group
is as large as it can be. In that case we call
the Galois group of
over
and denote it by
.
Suppose is a finite cover of
. An intermediate cover
between
and
is a cover of
by
, such that
, in such a way that the covering maps are compatible, in the sense that
is the composition of
and
. This sort of compatibilty condition will be implicitly assumed whenever we chain together multiple instances of the
notation. Two intermediate covers
are equivalent if they cover each other, in a fashion compatible with all the other covering maps, thus
and
. We then have the analogous Galois correspondence:
Theorem 4 (Fundamental theorem of covering spaces) Let
be a Galois covering.
- (i) If
is an intermediate cover betwen
and
, then
is a Galois extension of
, and
is a subgroup of
.
- (ii) Conversely, if
is a subgroup of
, then there is a intermediate cover
, unique up to equivalence, such that
.
- (iii) If
and
, then
if and only if
is a subgroup of
.
- (iv) If
, then
is a Galois cover of
if and only if
is a normal subgroup of
. In that case,
is isomorphic to the quotient group
.
Example 5 Let
, and let
be the
-fold cover of
with covering map
. Then
is a Galois cover of
, and
is isomorphic to the cyclic group
. The intermediate covers are (up to equivalence) of the form
with covering map
where
divides
; they are also Galois covers, with
isomorphic to
and
isomorphic to the multiples of
in
.
Given the strong similarity between the two theorems, it is natural to ask if there is some more concrete connection between Galois theory and the theory of finite covers.
In one direction, if the manifolds have an algebraic structure (or a complex structure), then one can relate covering spaces to field extensions by considering the field of rational functions (or meromorphic functions) on the space. For instance, if
and
is the coordinate on
, one can consider the field
of rational functions on
; the
-fold cover
with coordinate
from Example 5 similarly has a field
of rational functions. The covering
relates the two coordinates
by the relation
, at which point one sees that the rational functions
on
are a degree
extension of that of
(formed by adjoining the
root of unity
to
). In this way we see that Example 5 is in fact closely related to Example 3.
Exercise 6 What happens if one uses meromorphic functions in place of rational functions in the above example? (To answer this question, I found it convenient to use a discrete Fourier transform associated to the multiplicative action of the
roots of unity on
to decompose the meromorphic functions on
as a linear combination of functions invariant under this action, times a power
of the coordinate
for
.)
I was curious however about the reverse direction. Starting with some field extensions , is it is possible to create manifold like spaces
associated to these fields in such a fashion that (say)
behaves like a “covering space” to
with a group
of deck transformations isomorphic to
, so that the Galois correspondences agree? Also, given how the notion of a path (and associated concepts such as loops, monodromy and the fundamental group) play a prominent role in the theory of covering spaces, can spaces such as
or
also come with a notion of a path that is somehow compatible with the Galois correspondence?
The standard answer from modern algebraic geometry (as articulated for instance in this nice MathOverflow answer by Minhyong Kim) is to set equal to the spectrum
of the field
. As a set, the spectrum
of a commutative ring
is defined as the set of prime ideals of
. Generally speaking, the map
that maps a commutative ring to its spectrum tends to act like an inverse of the operation that maps a space
to a ring of functions on that space. For instance, if one considers the commutative ring
of regular functions on
, then each point
in
gives rise to the prime ideal
, and one can check that these are the only such prime ideals (other than the zero ideal
), giving an almost one-to-one correspondence between
and
. (The zero ideal corresponds instead to the generic point of
.)
Of course, the spectrum of a field such as is just a point, as the zero ideal
is the only prime ideal. Naively, it would then seem that there is not enough space inside such a point to support a rich enough structure of paths to recover the Galois theory of this field. In modern algebraic geometry, one addresses this issue by considering not just the set-theoretic elements of
, but more general “base points”
that map from some other (affine) scheme
to
(one could also consider non-affine base points of course). One has to rework many of the fundamentals of the subject to accommodate this “relative point of view“, for instance replacing the usual notion of topology with an étale topology, but once one does so one obtains a very satisfactory theory.
As an exercise, I set myself the task of trying to interpret Galois theory as an analogue of covering space theory in a more classical fashion, without explicit reference to more modern concepts such as schemes, spectra, or étale topology. After some experimentation, I found a reasonably satisfactory way to do so as follows. The space that one associates with
in this classical perspective is not the single point
, but instead the much larger space consisting of ring homomorphisms
from
to arbitrary integral domains
; informally,
consists of all the “models” or “representations” of
(in the spirit of this previous blog post). (There is a technical set-theoretic issue here because the class of integral domains
is a proper class, so that
will also be a proper class; I will completely ignore such technicalities in this post.) We view each such homomorphism
as a single point in
. The analogous notion of a path from one point
to another
is then a homomorphism
of integral domains, such that
is the composition of
with
. Note that every prime ideal
in the spectrum
of a commutative ring
gives rise to a point
in the space
defined here, namely the quotient map
to the ring
, which is an integral domain because
is prime. So one can think of
as being a distinguished subset of
; alternatively, one can think of
as a sort of “penumbra” surrounding
. In particular, when
is a field,
defines a special point
in
, namely the identity homomorphism
.
Below the fold I would like to record this interpretation of Galois theory, by first revisiting the theory of covering spaces using paths as the basic building block, and then adapting that theory to the theory of field extensions using the spaces indicated above. This is not too far from the usual scheme-theoretic way of phrasing the connection between the two topics (basically I have replaced étale-type points with more classical points
), but I had not seen it explicitly articulated before, so I am recording it here for my own benefit and for any other readers who may be interested.
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