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.


11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
Full Text: DOI Numdam EuDML


[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
[2] A. Fröhlich, M. J. Taylor, Algebraic Number Theory. Cambridge University Press, 1991. · Zbl 0744.11001
[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
[4] L. R. McCulloh, A Stickelberger condition on Galois module structure for Kummer extensions of prime degree. Dans Algebraic Number Fields Proceedings of the Durham Symposium 1975, Academic press, London, 1977. · Zbl 0389.12005
[5] H. B. Mann, On integral Basis. Proc. Amer. Math. Soc. 9 (1958), 119-149. · Zbl 0081.26602
[6] L.C. Washington, Introduction to Cyclotomic Fields. Springer-Verlag, New-York, 1982. · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.