## 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

### Keywords:

tame abelian extension; normal integral basis

### Citations:

Zbl 0941.11044; Zbl 1020.11070
Full Text:

### 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 [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
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