| Abstract |
Galois theory is an elegant interaction between field theory and group
theory. It gives a bijective correspondence between intermediate fields of
a Galois extension and subgroups of the Galois group of this extension.
For the case of finite Galois extensions, the fundamental theorem of Galois
theory establishes the bijective correspondence between intermediate
fields and subgroups. The fundamental theorem is useful in many situations
because it allows us to find out informations about the intermediate
fields of a Galois extension from the subgroups of the Galois group of the
extension, and vice versa. For this reason we would like to extend this
theorem to the case of infinite Galois extensions. In the first chapter we
study the Galois theory for infinite extensions. Luckily, the definition of
the Galois extension carries over without change from the finite case to
the case of infinite algebraic extensions. Unfortunately, the main theorem
doesn’t hold for infinite Galois extensions. This was ascertained by R.
Dedekind in 1897. But we can find out that there exists a fundamental
theorem for infinite Galois extensions which is a generalization of the
main theorem of the finite Galois theory. For its proof we are going to
put a topology on the infinite Galois groups, the so called Krull topology,
which we define. So the concept of “topological groups” will naturally
arise and for this reason we study them.
The groups which occur as Galois groups of field extensions belong to
a class of topological groups, the so-called profinite groups. This category
of groups we investigate in the second chapter. These groups are fairly
close relatives of finite groups. A profinite group is a topological group
that can be realized as a projective limit of finite topological groups. For
this reason we introduce the notion of projective limit. Also, we provide
some useful characterizations of profinite groups. One of them asserts
that a profinite group is a topological group which is Hausdorff, compact
and totally disconnected. But these are actually very familiar properties.
We have proved that a Galois group equipped with Krull topology has
these properties too. So we have that Galois groups are profinite groups.
In addition, we give some examples of profinite groups. Moreover, we
define the dual construction of projective limit, which is the direct limit,
and we prove some properties of projective and direct limits. For their
proof we need some notions of category theory, which we define.
In the last decades cohomology of groups has played a central role in
various branches of mathematics. Cohomology has a lot of applications
in class field theory and it has played an important role for its development.
In the third chapter we investigate the cohomology of finite groups.
Firstly, we define the differential groups because they serve as an introduction
to some of the basic techniques for studying the cohomology
groups. Also, we present some general considerations about G-modules.
In order to give the definition of cohomology group we introduce an
extensive formalism of homomorphisms, modules and sequence, the socalled
standard complex. Then we analyze the concrete meaning of the
cohomology group. As seen in the definition of group cohomology, it is
in general painful to find the nth cohomology group for an arbitrary finite
group G. We remark that in algebraic applications only the cohomology
groups of low dimension appear, since for these groups we have a concrete
algebraic interpretation. For this reason we study them completely.
Moreover, we study the cohomology of cyclic groups in which we prove
some essential statements of cohomology theory and we introduce the
Herbrand Quotient. We present also a lot of important theorems of cohomology
without their proof, such as Nakayama-Tate’s theorem which
is about the cohomological triviality. Another theorem is about the exactness
of the cohomology sequence. For its proof we need some special
mappings which we define.
In fourth chapter we study the cohomology of profinite groups. Their
cohomology groups often contain important arithmetic information. We
construct the cohomology group and for doing this we use the notion of
discrete modules. So firstly we define discrete G-modules and we provide
a characterization of them. Also, we calculate the cohomology groups in
low dimension. Moreover, we investigate what happens to the cohomology
groups Hq(G;A) if we change the group G, where A is a discrete
module. For doing this we need the notion of compatible pairs and some
properties of them. Finally, we study some special homomorphisms of
cohomology groups, such as the restiction and inflation, which they con-
nect the cohomology group of a group G with the cohomology group of
a subgroup of G.
All this theory played an essential role in number theory. So in the
last chapter we present the use of cohomology theory to solve problems
in number theory. The cohomology theory help us to think about
the extension problem of a group, since for an abelian group A which
is a G-module there is a natural bijective correspondence between the
equivalence classes of extensions of A by G and the elements of second
cohomology group H2. This is the reason why we define the extension
problem and we prove this correspondence. Moreover by the use of the
second cohomology group H2 we can define the Brauer group. We have
proved that Galois groups are profinite groups. A reasonable question is
if the converse is true. It is an important result that any finite group is
the Galois group of some field extension. This fact we can generalize to
profinite groups. More precisely we prove that every profinite group is
the Galois group of some field extension.
|
Profinite groups and cohomology
