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Swan modules and Hilbert-Speiser number fields. (English) Zbl 0941.11044

The paper gives necessary conditions for a number field \(K\) to have the Hilbert-Speiser property, from which it follows that \(\mathbb Q\) is the only such field. The property requires that for each tamely ramified abelian extension \(N\) of \(K\) the ring \({\mathfrak o}_N\) of integers in \(N\) has a normal integral basis over \({\mathfrak o}_K\). It implies, e.g., that the class number of \(K\) is 1 and that the exponent of the quotient \(({\mathfrak o}_K/l)^\times \) modulo the image of \({\mathfrak o}_K^\times\) divides \((l-1)^2/2\) for any odd prime \(l\). This last condition applies only to \(\mathbb Q\). The proof rests on a theorem of McCulloh which describes the set \(R\) of realizable classes in the class group \(Cl({\mathfrak o}_KG)\), i.e., of classes \([{\mathfrak o}_L]\) with \(L\) running through the tame Galois extensions of \(K\) with \(G(L/K)\simeq G\), a given elementary abelian group. When \(G\) is \(l\)-elementary abelian of order \(l^n\), this result is combined with \(T^{l^{n-1}(l-1)/2}\subset R\cap D\), where \(T\) is the Swan subgroup of \(Cl({\mathfrak O}_KG)\) and \(D\) the so-called kernel group. The containment results from a natural Mayer-Vietoris sequence by which \(T\) and \(({\mathfrak o}_K/l^n)^\times\) are related, and from the description of \(R\) in terms of the Stickelberger ideal. Special cases are already in D. R. Replogle [J. Algebra 212, 482–494 (1999; Zbl 0923.11152)]. Lower bounds for \(T\) then show the uniqueness of \(\mathbb Q\).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers

Citations:

Zbl 0923.11152
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References:

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