Hopf-Galois module structure of tame biquadratic extensions

Paul J. Truman

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James Carter (1998)

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A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic $p$

Nigel P. Byott (2011)

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Let $S / R$ be a finite extension of discrete valuation rings of characteristic $p > 0$ , and suppose that the corresponding extension $L / K$ of fields of fractions is separable and is $H$ -Galois for some $K$ -Hopf algebra $H$ . Let $𝔻_{S / R}$ be the different of $S / R$ . We show that if $S / R$ is totally ramified and its degree $n$ is a power of $p$ , then any element $ρ$ of $L$ with $v_{L} (ρ) \equiv - v_{L} (𝔻_{S / R}) - 1 (mod n)$ generates $L$ as an $H$ -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois...

Note on the Galois module structure of quadratic extensions

Günter Lettl (1994)

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In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, $ℚ^{(p)}$ . For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of $ℚ^{(n)} / ℚ^{{(n)}^{+}}$ .

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We establish automatic realizations of Galois groups among groups $M ⋊ G$ , where $G$ is a cyclic group of order $p^{n}$ for a prime $p$ and $M$ is a quotient of the group ring $𝔽_{p} [G]$ .

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PAC fields over number fields

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