Stickelberger ideals of conductor \(p\) and their application. (English) Zbl 1102.11059

Ichimura, Humio; Sumida-Takahashi, Hiroki

J. Math. Soc. Japan 58, No. 3, 885-902 (2006).

Authors’ abstract: Let \(p\) be an odd prime number and \(F\) a number field. Let \(K=F(\zeta_ p)\) and \(\Delta= \text{Gal}(K/F)\). Let \(S_\Delta\) be the Stickelberger ideal of the group ring \(Z[\Delta]\) defined in the previous paper [H. Ichimura, Stickelberger ideals and normal bases of rings of \(p\)-integers, Math. J. Okayama Univ., in press.]. As a consequence of a \(p\)-integer version of a theorem of L. R. McCulloh [A Stickelberger condition on Galois module structure for Kummer extensions of prime degree, Algebr. Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975, 561–588 (1977; Zbl 0389.12005); Galois module structure of elementary abelian extensions, J. Algebra 82, 102–134 (1983; Zbl 0508.12008)], it follows that \(F\) has the Hilbert-Speiser type property for the rings of p-integers of elementary abelian extensions over \(F\) of exponent \(p\) if and only if the ideal \(S_\Delta\) annihilates the \(p\)-ideal class group of \(K\). In this paper, we study some properties of the ideal \(S_\Delta\), and check whether or not a subfield of \(\mathbb Q(\zeta_p)\) satisfies the above property.

Reviewer: Maneesh Thakur (Bangalore)

MSC:

11R18	Cyclotomic extensions
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers

Keywords:

Stickelberger ideal; normal integral basis

Citations:

Zbl 0389.12005; Zbl 0508.12008

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