The branch of number theory with the basic aim
of studying properties of algebraic integers in algebraic number fields
of finite degree over the field
of rational numbers (cf.
Algebraic number).
The set of algebraic integers
of a field
— an
extension
of
of degree
(cf.
Extension of a field)
— can
be obtained from an integral basis
;
this means that each algebraic integer (i.e. element of
)
can be written in the form
where all the
run through the rational integers (i.e.
).
Moreover, such a representation is unique for each algebraic integer in
.
However, properties of rational integers often do not have obvious analogues
for algebraic integers. The first such property is
related to units, the invertible elements of
(cf.
Unit).
The field of rational numbers has only
and
as units, but a general algebraic number field may contain
an infinite number of units. E.g. consider the real quadratic field
,
where
is a rational integer not equal to a square. If, moreover,
,
then
is an integral basis for it. The
Pell equation
has an infinite number of solutions
.
Any of them gives rise to a unit
of
.
In fact,
and
is also an algebraic integer in
.
The units of
form an infinite multiplicative group (the
group of Pell units).
The question arises: What is the structure of this group?
A second property of rational numbers without an obvious analogue for
algebraic numbers is related to the theorem on unique factorization of rational integers
in prime factors:
For algebraic numbers such a unique factorization need not hold. E.g. consider the field
.
In it, the number 6 has two essentially different factorizations:
,
.
For extensions of higher degree the situation becomes still more complicated.
The question arises: What becomes of the unique factorization theorem, does
it have a meaning at all in algebraic number fields?
A third property without an obvious analogue is related to prime numbers. An
ordinary prime number need not remain a prime number
in an algebraic number field. E.g. in the field
of Gaussian numbers the prime number 5 has a factorization:
.
However, the prime number 7 remains prime in this field. The question
arises: Do there exist general laws governing the behaviour of prime numbers
in algebraic number fields of higher degree? In other words, is it
possible to find rules that would give a definite answer to the
following question: Does a given prime number remain prime in a field
or does it split in it, and if it splits, in how many factors?
Finally, a last (fourth) problem concerns the general
structure of algebraic number fields. The field
is the minimal field of characteristic zero without
proper subfields. Any other algebraic number field does have subfields. E.g.,
is a subfield of any algebraic number field. The question
arises: How many subfields are contained in a given extension
,
finitely or infinitely many, and what is their structure?
These are four main problems in algebraic number theory, and answering
them constitutes the content of algebraic number theory. It is quite natural to
start with the fourth problem, since its answer will shed light on
the other three. The problem was solved by
E. Galois
in the
1820s
(cf.
Galois theory).
The fact that the number of subfields of an extension
of degree
over
is finite follows from the existence of a one-to-one correspondence (the
fundamental Galois correspondence)
between all subfields of the normal closure of
(cf.
Normal extension)
and the subgroups of its Galois group, a finite
group of finite order (number of elements) at most
.
The structure of the group of units of a field was elucidated by
P. Dirichlet.
His
basic idea can be given by taking the group of Pell units (cf. above)
as example. Any power of such a unit (both positive or negative) is a unit. There is a
fundamental unit
,
and all other units are plus or minus integral
powers of it; hence, the Pell units form the product of
with an infinite cyclic group. This fact is a special case of the general
Dirichlet unit theorem for algebraic number fields:
If the degree of a field
is
,
where
is the number of real imbeddings
and
the number of complex-conjugate pairs of complex imbeddings
,
then the group of units
of
is the product of a finite cyclic group
and
infinite cyclic groups:
Here
are independent fundamental units and
is the group of roots of unity contained in
.
The norm of any unit in a field, i.e. the
product of the unit by its conjugates, is equal to
.
The problem of the non-unique factorization of algebraic integers in algebraic
number fields was solved by
E. Kummer,
who started from a
special case; he tried to solve Fermat's last
theorem on the impossibility of solving the equation
in non-zero integers for any prime number
.
Kummer expanded the left-hand side using
-th
roots of unity, and hence the problem was reduced to one about the algebraic integers of
,
a primitive
-th
root of unity. If there would be unique factorization into prime factors in
,
then it would have been sufficient to prove that not all prime factors
at the left-hand side occur with an exponent that is a multiple of
.
This was Kummer's first point of view, but Dirichlet pointed out to him
that unique factorization need not hold. In
order to overcome this difficulty Kummer introduced
ideal numbers,
thus altering the entire structure of algebraic number theory
for the future. The concept of an ideal number arises from the fact that if a field
does not contain prime numbers into which any algebraic integer in
can be split, then there is a field
of finite degree over
in which there does exist a collection of numbers that play the role of primes for
.
These numbers were called ideal by Kummer (since they do not lie in the original field
).
By introducing ideal numbers the theorem on unique factorization in
holds. Two numbers in a field that differ by a unit (so-called
associated numbers)
have one and the same ideal factors. (Note that
ideal numbers are defined relative to the original field
;
for another field
one must construct an extension
(of possibly different degree over
)
in which all ideal numbers of
are contained.)
Kummer also introduced the concept of the
ideal class number
of a field
.
Two ideal numbers are said to belong to the same class if their quotient belongs to
.
The number of classes thus obtained is called the ideal class number of
.
He obtained the following important result: The class number
of
is finite, and the classes form an Abelian group under
multiplication. Thus, any ideal number can be regarded as the
-th
root of some element of the original field
.
The class number can be explicitly described in
terms of other field constants (the regulator, the
discriminant and the degree of the field).
Subsequently, the concept of an ideal number was replaced by that of an
ideal,
equivalent to it, which can be described in terms of the field
itself. Already in the
1950s
the concept of an ideal
was generalized to the more comprehensive concept of a
divisor.
Therefore, the modern theory of Kummer may be stated in terms of divisors.
For algebraic number fields, however, the concepts of a divisor and an ideal
coincide, and such fields only will be considered in the sequel. The concept
of an ideal is related to that of non-associated numbers, which
helps in understanding the deep connections between Kummer's theory and Dirichlet's
theory of units. Although Kummer did not succeed in solving Fermat's problem,
his ideas extended far beyond this problem and the concept of
an ideal has now become fundamental in many branches of mathematics.
Since prime ideal numbers are defined relative to a field or, in modern
terminology, since they are prime ideals in a field, the third problem
on the factorization of ordinary prime numbers in an algebraic number field can
be stated in the following general form. Suppose one is given a field
and a prime ideal
of its ring of algebraic integers. The question is: Does
remain prime in an extension
or does it split in a product of prime ideals of the ring of algebraic integers of
?
In the latter case: By which law does it split? This question led to
class field theory,
a central part of modern algebraic number theory. The first solution
of this problem was given by Kummer, who proved that if
is a root of an irreducible polynomial
and
is an integral basis for
over
,
then
splits in
"by the same law as fx"
does in the residue field
(
).
In other words, the factorization of
in
is determined by the congruence
The corresponding factorization is called
Kummer's formula
(or
Kummer's decomposition)
where
are prime ideals in
.
In principle, this equation solves the third problem in algebraic number theory, but
it is a local equation in the sense that it is necessary to check it for each
prime ideal separately. The problem of partitioning the set
of all prime ideals in classes such that the factorization law
is the same in any given class and such that, moreover,
it is possible to give a simple description of these
classes, is solved by class field theory for extensions
with Abelian Galois group
.
Equation
(1)
yields a preliminary concept of a class. Let
be the degree of the extension
and let
be the relative degree of the ideal
.
Computing the relative norms
of both parts of
(1)
leads to
in which
and
are natural numbers. For
fixed equation
(2)
has a finite number of solutions,
so that the set of all prime ideals of
can be partitioned into a finite number of classes and one
can collect the prime ideals whose Kummer decomposition corresponds to one solution
of
(2)
into one class. The number of prime ideals with the property that some
,
is finite, and they all are divisors of the discriminant
of
.
Only infinite classes are of interest, so that classes for which
can be ignored.
In order to simplify the exposition, the field
is considered to be normal from now on. In such fields the condition
holds. Therefore, the set of all
not dividing
is partitioned into
classes, where
denotes the number of divisors. The class with
is of special interest, in it
A prime ideal
in it has maximal number of prime divisors in
:
Such
are said to
split completely
(or to be
totally split),
and their class is called the
principal class
of
relative to
.
It is a principal object of study in class field theory. In order
to be able to define the principal class via
(3)
it is necessary to
give a proof of the facts that prime ideals satisfying
(1)
do exist in
and that there are infinitely many such ideals. Therefore a basic problem in class
field theory is to define the principal class in terms of the field
itself in such a way that its infinite nature would
follow. This problem has been completely solved for Abelian extensions
.
In order to be able to understand the ideas of class field theory better,
it is necessary to introduce the general concept of an ideal class group.
Kummer's definition given above then corresponds to the modern concept of an
absolute ideal class group. The general concept of an ideal class group
which is used nowadays is due to
H. Weber
and
T. Takagi,
[5].
Weber introduced the concept of the
conductor of a class group.
Let
be an integral ideal of
,
let
be the subgroup of principal ideals
of
given by
and let
be the subgroup of all ideals of
that are relatively prime with
.
A subgroup
satisfying
can be regarded as a group of principal ideals, and a (generalized) class group
is constructed in this way. For
and
one obtains Kummer's definition. In the general case
consists of progressions
whose residue classes form a subgroup of the whole multiplicative group
.
The order
of this generalized class group satisfies the estimates
,
where
is the order of the absolute class group and
is Euler's phi-function. For different conductors
and
the class groups may be equivalent if
and
consists in fact of the same progressions
,
where
.
If one agrees not to distinguish between equivalent class
groups, then one obtains the concept of a
class group according to Weber
with a conductor
that is the greatest common divisor of all equivalent conductors. A
class field according to Weber
is a field
in which only the prime ideals in its principal class
,
and only these, split completely. Dirichlet's theorem on prime
ideals in progressions, which is valid in every field
,
implies that there are infinitely many prime ideals.
Weber has shown that in certain particular cases the Galois group
of a class field and the class group
of
are isomorphic.
A new point of view on class field theory, which is still
valid, originated with
D. Hilbert.
He understood that
between all relatively Abelian extensions of a field
and all class fields for this field there must exist
a one-to-one correspondence. This correspondence can be
stated as follows. If for some conductor
one constructs the Weber class group, then there is only one normal extension
in which the prime ideals of Weber's principal
class, and only these, will split completely. Moreover, the Galois group of
is isomorphic to the Weber class group and the discriminant
of
consists of the same prime ideals as the conductor
.
The converse is also true: If an Abelian extension with Galois group
is given, then there exists a method (subsequently explicitly
formulated by Takagi) by which one can uniquely construct a principal class
such that the class group
is isomorphic to
,
only the prime ideals of
split completely in
,
and such that the conductor
has the same prime divisors as the discriminant
of
.
The
"duality"
was stated by Hilbert in
1900
as a hypothesis (he proved it
only in special cases). In its general form it was proved by Takagi.
The next important stage of development of class group theory is related to
the name of
E. Artin,
who revealed the special role of the
canonical isomorphism between the Galois group and the ideal class
group. He proved that this isomorphism is given by the
Frobenius automorphism
of an Abelian extension
,
defined as
Here
is the absolute norm of the ideal
,
runs through all numbers of
and
is a prime ideal of
.
The automorphism
(nowadays denoted by
depends (in an appropriate group) only on the ideal class to which
belongs. It is multiplicative:
where the symbol at the right-hand side is understood
as an automorphism of the class to which
belongs. With this in mind, Artin introduced the symbol
on the entire group
of ideals
of
that are relatively prime with the conductor
.
It is called the
Artin symbol
and realizes an isomorphism between the class group
and the Galois group
,
whose explicit from is expressed in
Artin's reciprocity law:
if and only if
(reciprocity as a correspondence between the groups
and
).
From this one can obtain the classical form of the
reciprocity law
in the language of Kummer's power reciprocity symbol (one has to consider the field
,
cf.
Reciprocity laws).
This form, in turn, implies the reciprocity law for Hilbert's
symbol. In all three forms the reciprocity law is regarded as an
isomorphism of groups, and the symbols of Artin, Kummer and Hilbert
are regarded as group elements realizing this isomorphism. However, each of
them also has a numerical value, which is equal to some
-th
root of unity. Therefore one can formulate the
reciprocity law as follows. Given the value of
,
it is required to find the value of the symbol
reciprocal to it, i.e. to exhibit the explicit form of the function of
defined by
In this form the law first appeared in the work of
C.F. Gauss
(cf.
Gauss reciprocity law)
for quadratic fields and in the work of Kummer for cyclotomic fields
of prime degree. The study of the reciprocity law in this
form was subsequently continued by
C.G.J. Jacobi,
F.G.M. Eisenstein,
Hilbert,
H. Hasse,
and others, but they only obtained partial results.
I.R. Shafarevich
[6]
solved this problem in its general form in
1948,
basing himself on
the idea of establishing a connection between the arithmetic definition of the symbol
and the analytic definition of an Abelian differential
on a Riemann surface. He gave a construction of
which precisely corresponds to the determination of the residue of
in the
-adic
field. For this purpose he introduced
-functions,
also called
Shafarevich functions,
in terms of which he also obtained an explicit
form of the reciprocity law in the general case.
In the late
1920s
Hasse introduced the idea of
doing class field theory for one prime ideal of
at a time, and reformulated and proved many theorems for an Abelian extension
of a local field
.
At first the idea was of secondary importance only (the results were
obtained as a consequence of the local theory), but at
the end of the
1930s
C. Chevalley
proved the extraordinary importance of the
local point of view in the creation of class field theory. He introduced
the concept of an idèle group and formulated general class field
theory from the local point of view. Following this the
local-global principle
became established in class field theory. Subsequently it was extended and refined (cf.
[5]),
and as a result Abelian class field theory took a structured
and completed form. The problem of creating non-Abelian class field
theory for normal extensions with non-Abelian Galois group remains.
The exposition above relates mainly to the qualitative aspects of
algebraic number theory. In questions of quantitative estimation and methods
algebraic number theory is intimately connected with analytic number theory. It
is also based, to a large extent, on properties of
-functions
and
-functions
of algebraic fields.