On a certain subideal of the Stickelberger ideal of a cyclotomic field. (English) Zbl 0629.12005

Kučera, Radan

Arch. Math., Brno 22, 7-20 (1986).

Consider the group ring \(R={\mathbb Z}[G]\) of the Galois group of the \(m\)th cyclotomic field \(\mathbb Q(\xi_ m)\) over the ring \(\mathbb Z\) of rational integers (\(m\in\mathbb Z\), \(m>2\), \(m\not\equiv 2\pmod 4\), \(\xi_ m=e^{2\pi i/m}\)) and the subring \(R^-=(1-j)R\), where \(j\in G\) means the complex conjugation. In the ring \(R\) a special ideal \(S\), called the Stickelberger ideal, is defined and often used. For this \(m\) W. Sinnott [Ann. Math. (2) 108, 107–134 (1978; Zbl 0395.12014)] generalized K. Iwasawa’s class number formula [ibid. 76, 171–179 (1962; Zbl 0125.020)] \((m=p^{n+1}\), \(p\) an odd prime, \(n\geq 0\)) by showing finiteness of the quotient group \(R^-/S^-\) and computing the group index \([R^- :S^-]\) \((S^-=S\cap R^-).\)
In this paper the ideal \(I\) [considered by L. C. Washington in his book “Introduction to cyclotomic fields”, Grad. Texts Math. 83 (1982; Zbl 0484.12001)], smaller than \(S\), is defined and a special basis of \(I^-=I\cap R^-\) (as a \(\mathbb Z\)-module) is determined. By means of the calculation of the determinant of the transition matrix from a suitable basis of the \(\mathbb Z\)-module \(R^-\) to the introduced basis of \(I^-\) the group index \([R^- : I^-]\) is determined. The equality \(I^-=S^-\) is valid if and only if \(m=p^{n+1}\) as in Iwasawa’s case (4.3). But in many cases (determined in 4.1) the group index \([R^- : I^-]\) is even infinite.

Reviewer: L.Skula

MSC:

11R18	Cyclotomic extensions
11R23	Iwasawa theory
11R52	Quaternion and other division algebras: arithmetic, zeta functions

Keywords:

basis of Z-module; cyclotomic field; Stickelberger ideal; class number formula; group index

Citations:

Zbl 0395.12014; Zbl 0125.020; Zbl 0484.12001