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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
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