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 -adic -functions" , Ann. Math. Studies , 74 , Princeton Univ. Press (1972)
[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)