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ON COUNTING RINGS OF INTEGERS AS GALOIS MODULES Abstract. Let K be a number field and G a finite abelian group. We study the asymptotic
 

Summary: ON COUNTING RINGS OF INTEGERS AS GALOIS MODULES
A. AGBOOLA
Abstract. Let K be a number field and G a finite abelian group. We study the asymptotic
behaviour of the number of tamely ramified G­extensions of K with ring of integers of fixed
realisable class as a Galois module.
1. Introduction
Suppose that K is a number field with ring of integers OK , and let G be a fixed, finite
group. If K h /K is a tamely ramified Galois algebra with Galois group G, then a classical
theorem of E. N˜oether implies that the ring of integers O h of K h is a locally free OKG­
module. It therefore determines a class (O h ) in the locally free class group Cl(OK G) of
OKG. We say that a class c # Cl(OK G) is realisable if c = (O h ) for some tamely ramified
G­algebra K h /K, and we write R(OKG) for the set of realisable classes in Cl(OK G). These
classes are natural objects of study, and they arise, for instance, in the context of obtaining
explicit analogues of known Adams­Riemann­Roch theorems for locally free class groups (see
e.g. [1, §4] and the references cited there; see also the work of B. K˜ock ([4], [5]) on this and
related topics). We also remark that the problem of describing R(OKG) for arbitrary finite
groups G may be viewed as being a Galois module theoretic analogue of the inverse Galois
problem for finite groups.
When G is abelian, Leon McCulloh has obtained a complete description of R(OKG) in
terms of certain Stickelberger homomorphisms on classgroups (see [7]). In particular, he has

  

Source: Agboola, Adebisi - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics