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On Hilbert-Speiser fields. (Sur les corps de Hilbert-Speiser.) (French) Zbl 1161.11398

Summary: A number field is called a Hilbert-Speiser field for a prime number \(p\) if each tamely ramified finite abelian extension of degree \(p\) admits a normal integral basis. A number field is called a Hilbert-Speiser field if it’s Hilbert-Speiser for all primes \(p\). It’s well known that \(\mathbb{Q}\) is such a field. In the article [J. Number Theory 79, No. 1, 164–173 (1999; Zbl 0941.11044)], C. Greither, D. R. Replogle, K. Rubin and A. Srivastav showed that \(\mathbb{Q}\) is the only Hilbert-Speiser field. We give here a necessary and sufficient condition for a field to be Hilbert-Speiser for \(p=2\). For example \(\mathbb{Q}(\sqrt{p})\) is a Hilbert-Speiser field for \(p=2\) if and only if its class number is one. Then generalizing works of M. Conrad and D. R. Replogle [Math. Comput. 72, No. 242, 891–899 (2003; Zbl 1020.11070)] we obtain prime numbers \(p\) for which an imaginary abelian field is a Hilbert-Speiser field for \(p\), and we also give a criterion for real abelian fields.

MSC:

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

[1] M. Conrad, D. R. Replogle, Nontrivial Galois Module Structure of cycloyomic Fields. Mathematic of computation 72 (2003), no. 242, 891-899. · Zbl 1020.11070
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[3] C. Greither, D. R. Replogle, K. Rubin, A. Srivastav, Swan Modules and Hilbert-Speiser number fields. J. of Number theory 79 (1999), 164-173. · Zbl 0941.11044
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