On normal integral bases in ray class fields over imaginary quadratic fields. (English) Zbl 0866.11064

Greither, Cornelius

Acta Arith. 78, No. 4, 315-329 (1997).

The question which motivated this paper is the following: Given an imaginary quadratic field \(K\) and a prime ideal \({\mathfrak p}\) of \(K\), does the ray class field \(K({\mathfrak p})\) have normal integral basis over \(K(1)\), the Hilbert class field? This is an instance of the “relative tame problem in Galois module structure”. Thanks to a counterexample of E. J. Gómez Ayala and R. Schertz [J. Number Theory 44, 41-46 (1993; Zbl 0777.11046)] using quadratic subextensions, one knows the answer to the problem is no; however this seemed to be due to a peculiarity of the number 2, as witnessed by positive results of E. J. Gómez Ayala [Acta Arith. 72, 375-383 (1995; Zbl 0857.11060)] for cases where the degree of the extension is three. Thus a positive answer, under suitable restrictions or modifications, seemed a possibility.
In the present paper it is shown, however, that the answer to the initial question is a solid no in general, and remains so even after relaxing somewhat the notion of a normal integral basis. A systematic way of finding counterexamples is presented. This involves some descent theory, manipulation of Stickelberger ideals, and some explicit calculations in class groups, which were done by the program package PARI.

Reviewer: C.Greither (Sainte-Foy / PQ)

MSC:

11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R37	Class field theory
11R29	Class numbers, class groups, discriminants
11Y40	Algebraic number theory computations

Keywords:

normal bases; class groups; cyclotomic descent; prime ideal; ray class field; normal integral basis; Stickelberger ideals; PARI

Citations:

Zbl 0777.11046; Zbl 0857.11060