Stickelberger ideal
Let be a fixed natural number and
the cyclotomic field generated by a primitive
th root of unity over the rational number field
. Let
be its Galois group and
the group ring of
(cf. also Group algebra; Cross product) over the rational integers
. They act on ideals and on the ideal class group
of
(cf. Class field theory). The Stickelberger ideal
is an ideal in
annihilating
and related with the relative class number
of
. It is defined as follows.
Let be the ring of integers of
and
a prime ideal of
that is prime to
. Let
be a prime integer satisfying
and let
be the prime field. Define a mapping
by
![]() |
where is the complex number field. Let
be the residue field and define
by composition of
and the trace mapping
, i.e.,
. Let
be the group of unities in
. Then there is an injection
![]() |
so that divides
with
. This induces a bijection
. Define a character
of order
by
![]() |
for . One can now make up the Gauss sum
![]() |
for . For a real number
, let
be the number uniquely determined by
and
. For
with
, let
be the element of
corresponding to
by
. For
, let
be the element
![]() |
L. Stickelberger proved the following theorem: For ,
one has
if and only if
. In this case, for any prime ideal
prime to
one has
and
is the principal ideal generated by
.
The ideal in
is called the Stickelberger ideal for
, and an element of
is called a Stickelberger operator for
. Since any class of
contains a prime ideal
prime to
, Stickelberger's theorem implies that
annihilates
:
.
For a -module
, one defines
, where
corresponds to complex conjugation.
Let be the number of primes that ramify in
. Now,
1) if , then
, where
is the relative class number of
(Iwasawa's theorem);
2) if , then
, where
(Sinnott's theorem).
These results have, to some extent, been generalized to absolute Abelian fields. Stickelberger ideals are also used to construct a -adic
-function [a3].
References
[a1] | L. Stickelberger, "Über eine Verallgemeinerung der Kreistheilung" Math. Ann. , 37 (1890) pp. 321–367 |
[a2] | K. Iwasawa, "A class number formula for cyclotomic fields" Ann. of Math. (2) , 76 (1962) pp. 171–179 |
[a3] | K. Iwasawa, "Lectures on ![]() ![]() |
[a4] | W. Sinnott, "On the Stickelberger ideal and the circular units of a cyclotomic field" Ann. of Math. (2) , 108 (1978) pp. 107–134 |
[a5] | W. Sinnott, "On the Stickelberger ideal and the circular units of an abelian field" Invent. Math. , 62 (1980/1) pp. 181–234 |
[a6] | T. Kimura, K. Horie, "On the Stickelberger ideal and the relative class number" Trans. Amer. Math. Soc. , 302 (1987) pp. 727–739 |
[a7] | S. Lang, "Cyclotomic fields" , 1–2 , Springer (1990) |
[a8] | L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) |
Stickelberger ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stickelberger_ideal&oldid=48836