Normal integral bases and the Spiegelungssatz of Scholz (original) (raw)

Abstract

Normal integral bases and the Spiegelungssatz of Scholz by Jan Brinkhuis (Rotterdam) Introduction. In Hilbert's Zahlbericht one can find the first global result on the Galois module structure of rings of integers ([H], Satz 132). Slightly extended it states that if N/Q is a tame extension with abelian Galois group ∆, then o N , the ring of integers in N , is free as a module over the group ring Z∆; moreover, an explicit canonical algebraic integer a such that o N = Z∆a can be given. The numbers a δ (δ ∈ ∆), the algebraic conjugates of a, are said to form a normal integral basis of the field extension N/Q. This result has been the starting point for a modern development, which has led to the deep result that the structure of o N as a module over Z∆, for arbitrary tame Galois extensions of number fields N/K with Galois group ∆, is determined in terms of the symplectic root numbers of N/K (see [F] and [T1]). Fröhlich's book [F] also contains a detailed introduction and a rather complete list of references. For a more recent survey on Galois module theory we refer to [Ca-Ch-F-T]. In the special case that ∆ is of odd order the result mentioned above gives that o N is a free Z∆-module, but the proof does not provide an explicit basis. If one considers the richer o K ∆-module structure of o N rather than the Z∆-module structure alone, then o N is expected to be "usually" not even free if K = Q. Results of Taylor show that by modifying both o N and o K ∆ one can sometimes-if K and N are certain ray class fields over imaginary quadratic number fields-achieve the "ideal" of free modules with explicit generators (see [C-T]). However, if one decides not to modify the original classical problem of the determination of the o K ∆-module structure of o N , then a natural question is to what extent, for given K and ∆, the realization of ∆ as a Galois group of a tame extension N/K is determined by the ramification of N/K together with the structure of o N as an o K ∆-module. This point of view is worked out in [B2]. In a sense the core of the question is how rare unramified extensions which possess a normal integral basis are. We mention in passing that this question is equivalent to a special case of a problem considered by Taylor in recent work, that of determining the kernel [1] This work was begun in May 1988 while the author enjoyed the hospitality of the University of Bordeaux I.

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