Cyclotomic field
A field obtained from the field
of rational numbers by adjoining a primitive
-th root of unity
, where
is a natural number. The term (local) cyclotomic field is also sometimes applied to the fields
, where
is the field of rational
-adic numbers. Since
when
is odd, it is usually assumed that
(
). Distinct
then correspond to non-isomorphic fields
.
Cyclotomic fields arise naturally in the cyclotomy problem — the division of a circle into equal parts is equivalent to the construction of a primitive root
in the complex plane. The structure of cyclotomic fields is "fairly simple" , and they therefore provide convenient experimental material in formulating general concepts in number theory. For example, the concept of an algebraic integer and a divisor first arose in the study of cyclotomic fields.
The special position of cyclotomic fields among all algebraic number fields is illustrated by the Kronecker–Weber theorem, which states that a finite extension is Abelian if and only if
for some
. An analogous proposition holds for local cyclotomic fields.
Algebraic theory.
The field is an Abelian extension of
with Galois group
![]() |
where is the multiplicative group of the ring of residues modulo
. In particular, the degree
is
, where
is Euler's function. The field
is totally imaginary and of degree 2 over its maximal totally real subfield
.
If is the factorization of
into prime numbers, then
is the linearly disjoint compositum of the fields
. In the field
the prime divisor
has ramification index
. In the same field one has the following equality of principal divisors:
. All other prime divisors of
are unramified in
, whence it follows that
is ramified in
if and only if
.
The numbers form an integral basis for the field
. The discriminant of
is equal to
. If
and
are fields which are linearly disjoint over
with relatively prime discriminants
and
, then
, where
,
. This makes it possible to calculate
for arbitrary
(see [3]).
For the field , the numbers
![]() |
where , generate a subgroup of finite index in the group of all units. The elements of this subgroup are known as circular units or cyclotomic units.
The decomposition law for cyclotomic fields, that is, the law according to which the prime divisors in
factorize into prime divisors in
, is a particular case of the general decomposition law in Abelian extensions, established in class field theory (see [4]). Explicitly: If
and if
is the least natural number such that
(
), then in
,
![]() |
where the prime divisors are pairwise distinct,
and
. Thus, the factorization type of
depends only on the residue of
(
). If
, the exact form of the factorization of
can be obtained, using the facts that
, where
and
is totally ramified in
.
If is the maximal Abelian extension of
, then
and
![]() |
where is the completion of the ring of integers
with respect to all ideals of finite index. In particular, for any prime number
there is a unique extension
with Galois group isomorphic to the group of
-adic integers
.
According to class field theory, there exists a reciprocity map
![]() |
where is the idèle group of
. In the case of a cyclotomic field
admits a simple explicit description (see [4]).
Analytic theory.
Many results regarding the structure of the divisor class group of can be proved by analytic methods. If
is the class number of
, then
![]() |
here and
are, respectively, the number of roots of unity, the discriminant and the regulator of
,
, and
is the Dirichlet
-function for the character
, where
runs through all non-trivial primitive multiplicative characters modulo
. The function
in turn can be expressed explicitly in terms of Gauss sums (see [7]). This solves the problem of calculating
, given
.
There is a natural decomposition of into two factors,
; the first and second factor of the class number, where
is interpreted as the class number of the field
. If
then
, where
is the group of units of
and
is the group of real cyclotomic units (any cyclotomic unit becomes real if multiplied by a suitable root of unity).
In questions related to the Fermat problem, an important role is played by the divisibility of the class number of by
, where
is prime. It is known that
(
) for infinitely many prime numbers
(such numbers
are said to be irregular). As to the set of regular prime numbers
, i.e. numbers for which
(
), it is not known (1982) whether it is finite or infinite. It has been conjectured that
(
) for all
, and this has been confirmed in a large number of cases. The factor
is more amenable to investigation. There exists a relatively simple criterion for the divisibility of
(and
) by
in terms of Bernoulli numbers ([7]). It is known that
as
and that
if and only if
(see [6]).
The so-called -adic
-functions have been successfully applied to the study of the class groups of cyclotomic fields (see [5], [8]).
References
[1] | E. Kummer, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten komplexen Zahlen in ihre Primfaktoren" J. Reine Angew. Math. , 35 (1847) pp. 327–367 |
[2] | H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) |
[3] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) |
[4] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) |
[5] | I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian) |
[6] | K. Uchida, "Class numbers of imaginary abelian number fields III" Tôhoku Math. J. , 23 (1971) pp. 573–580 |
[7] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966) |
[8] | K. Iwasawa, "Lectures on ![]() ![]() |
[9] | S. Lang, "Cyclotomic fields" , Springer (1978) |
Comments
It is still not known whether there are infinitely many regular primes (1987), but it is conjectured that the density of the set of regular primes inside the set of all primes is , see [a5]. It has been proved (for
) that
if and only if
or
or
, see [a3], [a5]. An important theorem that describes the structure of the class groups of cyclotomic fields in terms of
-adic
-functions has been proved by B. Mazur and A. Wiles (see [a1], [a4]) with the help of methods from algebraic geometry.
References
[a1] | J. Coates, "The work of Mazur and Wiles on cyclotomic fields." Sem. Bourbaki , 33 : 575 (1980/81) Lecture Notes in Math. , 901 (1981) pp. 220–242 |
[a2] | S. Lang, "Cyclotomic fields" , II , Springer (1980) |
[a3] | J.M. Masley, H.L. Montgomery, "Cyclotomic fields with unique factorization" J. Reine Angew. Math. , 286–287 (1976) pp. 248–256 |
[a4] | B. Mazur, A. Wiles, "Class fields of abelian extensions of ![]() |
[a5] | L.C. Washington, "Cyclotomic fields" , Springer (1982) |
Cyclotomic field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_field&oldid=23805