Totally Ramified Valuations on Finite-Dimensional Division Algebras

Tignol, J.-P.; Wadsworth, Adrian R.

doi:10.2307/2000907

Cited by 17 publications

(27 citation statements)

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“…We show that H 2 (G, µ) is generated by cup products, a well known result (see [TW,Remark 4.11]). We then develop a complete, computationally effective theory for the bottom row, for any proabelian group, using the theory of alternating forms.…”

Section: Introductionsupporting

confidence: 55%

Alternating forms and the Brauer group of a geometric field

Brussel

2007

Trans. Amer. Math. Soc.

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Section: Introductionsupporting

confidence: 55%

Alternating forms and the Brauer group of a geometric field

Brussel

2007

Trans. Amer. Math. Soc.

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“…Next, we recall the notion of armature ( [13]) which is basically the same as the notion of an abelian projective basis: Definition 8. Let A be a finite-dimensional F -algebra.…”

Section: Rigiditymentioning

confidence: 99%

Projective bases of division algebras and groups of central type II

Natapov

2008

Isr. J. Math.

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Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k α G for some α ∈ H 2 (G, k × ), where the action of G on k × is trivial. In a preceding paper by Aljadeff, Haile and the author it was shown that if a group G is a projective basis in a k-central division algebra then G is nilpotent and every Sylow p-subgroup of G is on the short list of p-groups, denoted by Λ. In this paper we complete the classification of projective bases of division algebras by showing that every group on that list is a projective basis for a suitable division algebra.We also consider the question of uniqueness of a projective basis of a k-central division algebra. We show that basically all groups on the list Λ but one satisfy certain rigidity property.

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“…Set Tk = 4t(zf», zf ; F') ®F, APk(zf , zf ; F') ®F,-*F. A^z^ , z%F>), and let T -T2 ®f ■ ■ ■ ®f' Tt. Then with respect to the (z¡ , ... , z2^)-adic valuation on F', we see that T is a totally ramified division algebra over F' of index m/pßl by [JW,Corollary 2.6], and T has exponent n/p"] by [TW,Theorem 4.7(i) that L is a maximal subfield of T. Since T is totally ramified over F', so is L; hence L/F' is Galois [TW,Proposition 1.4(iii)]. Thus N is normal in G. Since gcd(|7V| , \G/N\) = 1, the group G is a semidirect product N » 77 by the Schur-Zassenhaus theorem [R, p. 149]; in particular, G contains a subgroup 77 with \G:H\= pß'.…”

Section: The Examplesmentioning

confidence: 99%