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On the restricted Hilbert-Speiser and Leopoldt properties. (English) Zbl 1286.11186

Summary: Let \(G\) be a finite abelian group. A number field \(K\) is called a Hilbert-Speiser field of type \(G\) if, for every tame \(G\)-Galois extension \(L/K\), the ring of integers \(\mathcal O_L\) is free as an \(\mathcal O_K[G]\)-module. If \(\mathcal O_L\) is free over the associated order \(\mathcal A_{L/K}\) for every \(G\)-Galois extension \(L/K\), then \(K\) is called a Leopoldt field of type \(G\). It is well known (and easy to see) that if \(K\) is Leopoldt of type \(G\), then \(K\) is Hilbert-Speiser of type \(G\). We show that the converse does not hold in general, but that a modified version does hold for many number fields \(K\) (in particular, for \(K/\mathbb Q\) Galois) when \(G = C_{p}\) has prime order. We give examples with \(G = C_{5}\) to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants

Software:

QaoS; Magma; PARI/GP
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Full Text: arXiv Euclid

References:

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