## 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

Zbl 0923.11152
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### References:

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