## Hilbert-Speiser number fields and the complex conjugation.(English)Zbl 1247.11142

Let $$N/F$$ be a finite Galois extension of number fields with Galois group $$G$$. Noether showed that the ring of integers $$\mathcal{O}_{N}$$ is projective over the group ring $$\mathcal{O}_{F}[G]$$ if and only if $$N/F$$ is tame. The extension $$N/F$$ is said to have a normal integral basis if $$\mathcal{O}_{N}$$ is in fact free over $$\mathcal{O}_{F}[G]$$. The classical Hilbert-Speiser Theorem says that every finite tame abelian extension of the rationals $$\mathbb{Q}$$ has a normal integral basis.
Let $$p$$ be a prime number. A number field $$F$$ satisfies the condition $$(H_{p})$$ if every tame cyclic extension $$N/F$$ of degree $$p$$ has a normal integral basis. So the Hilbert-Speiser Theorem implies that $$\mathbb{Q}$$ satisfies $$(H_{p})$$ for every prime number $$p$$. On the other hand, C. Greither, D. R. Replogle, K. Rubin and A. Srivastav [J. Number Theory 79, No. 1, 164–173 (1999; Zbl 0941.11044)] showed that any number field $$F \neq \mathbb{Q}$$ does not satisfy $$(H_{p})$$ for infinitely many $$p$$.
The main result of the paper under review is as follows. Let $$p$$ be a prime number with $$p \geq 5$$ and let $$F/\mathbb{Q}$$ be a nonquadratic Galois extension such that $$F$$ is a CM field. Then $$F$$ satisfies the condition $$(H_{p})$$ if and only if $$F=\mathbb{Q}(\zeta_{12})$$ and $$p=5$$ (here $$\zeta_{12}$$ denotes a primitive $$12$$th root of unity). Combined with the result of H. Ichimura and H. Sumida-Takahashi [Acta Arith. 136, No. 4, 385–389 (2009; Zbl 1168.11047)] it follows that there exits no CM Galois extension of $$\mathbb{Q}$$ satisfying $$(H_{p})$$ for any $$p \geq 11$$. In the final section of the paper under review, it is shown that the real quadratic field $$F=\mathbb{Q}(\sqrt{5})$$ satisfies $$(H_{13})$$ and that the real cyclic cubic field $$F=\mathbb{Q}(\cos(2\pi/7))$$ satisfies $$(H_{11})$$.

### MSC:

 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R18 Cyclotomic extensions

### Citations:

Zbl 0941.11044; Zbl 1168.11047
Full Text:

### References:

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