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Hopf-Galois module structure of tame biquadratic extensions. (English. French summary) Zbl 1262.11095

Author’s abstract: In [New York J. Math. 17, 799–810 (2011; Zbl 1250.11098)] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields \(L/K\) with group \(G\propto C_2\times C_2\) and study in detail the local and global structure of the ring of integers \(\mathfrak O_L\) as a module over its associated order \(\mathfrak A_H\) in each of the Hopf algebras \(H\) giving a nonclassical Hopf-Galois structure on the extension. The results of [loc. cit.] imply that \(\mathfrak O_L\) is locally free over each \(\mathfrak A_H\), and we derive necessary and sufficient conditions for \(\mathfrak O_L\) to be free over each \(\mathfrak A_H\). In particular, we consider the case \(K=\mathbb Q\), and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.

MSC:

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

Citations:

Zbl 1250.11098
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References:

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