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

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.

### 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

### Citations:

Zbl 0777.11046; Zbl 0857.11060
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