Exact Sequences and Galois Module Structure

Chinburg, Ted

doi:10.2307/1971177

Cited by 64 publications

(70 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Chinburg [4] has proved this in the case when v is at most tamely ramified. We use Lemma 3.1 to reduce the proof in the wild case to the tame case.…”

Section: Proof Of Main Theoremmentioning

confidence: 87%

“…Chinburg [4] showed that 0(NÂK, 2) does not depend on the choices made in the definition and is thus an invariant of NÂK.…”

Section: Chinburg's Invariantmentioning

confidence: 99%

“…However W NÂ K is defined for wild extensions. Chinburg [4,5] asked whether 0(NÂK, 2)=W NÂK for arbitrary extensions. In support of this, Kim [9] has shown that for certain wildly ramified extensions of Q where the Galois group is the quaternion group of order 8, one has 0(NÂQ, 2)=W NÂ Q .…”

mentioning

confidence: 99%

See 2 more Smart Citations

On a Multiplicative–Additive Galois Invariant and Wildly Ramified Extensions

Rogers

1996

Journal of Number Theory

View full text Add to dashboard Cite

We investigate the Galois module structure of wildly ramified extensions. We are interested in particular in the second invariant of an extension of number fields defined by Chinburg via the canonical class of the extension and lying in the locally free class group. We show that in Queyrut's S-class group, where S is a (finite) set of primes, the image of Chinburg's invariant equals the stable isomorphism class of the ring of integers and thus extend Chinburg's result for tame extensions. 1996Academic Press, Inc.

show abstract

“…Chinburg [4] has proved this in the case when v is at most tamely ramified. We use Lemma 3.1 to reduce the proof in the wild case to the tame case.…”

Section: Proof Of Main Theoremmentioning

confidence: 87%

“…Chinburg [4] showed that 0(NÂK, 2) does not depend on the choices made in the definition and is thus an invariant of NÂK.…”

Section: Chinburg's Invariantmentioning

confidence: 99%

See 1 more Smart Citation

On a Multiplicative–Additive Galois Invariant and Wildly Ramified Extensions

Rogers

1996

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

“…The study of the locally free class group cl(Z[G]) has been to a very large extent motivated by questions and conjectures arising in the field of Galois module theory. Among the large amount of existing work we mention Fröhlich's conjecture (proved by Martin Taylor [19]) on the Galois module structure of rings of integers in tame Galois extensions of number fields and the Ω-conjectures of Chinburg [9], extending and generalizing Fröhlich's conjecture.…”

Section: Introductionmentioning

confidence: 99%

Computations in Relative Algebraic K-Groups

Bley

Wilson

2009

LMS J. Comput. Math.

View full text Add to dashboard Cite

Let G be finite group and K a number field or a p-adic field with ring of integers O K . In the first part of the manuscript we present an algorithm that computes the relative algebraicas an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in, K) and in the locally free class group cl(O K [G]). All algorithms have been implemented in Magma for the case K = Q.In the second part of the manuscript we prove formulae for the torsion subgroup of K 0 (Z[G], Q) for large classes of dihedral and quaternion groups.

show abstract

“…Since Chinburg's conjecture predicts that Ω m = 0 whenever G has no irreducible symplectic representation (cf. §3 of [3]), an envelope ω ω ω of µ µ µ with [ω ω ω] − w[ZG] = 0 and w = d(G) (cf. [9]) is a useful ingredient for examples.…”

Section: Discussionmentioning

confidence: 99%

Galois structure ofS-units

Pacheco

Weiss

2016

Bull. London Math. Soc.

View full text Add to dashboard Cite

Let K/k be a finite Galois extension of number fields with Galois group G, S a large G-stable set of primes of K, and E (respectively µ µ µ) the G-module of S-units of K, (resp. roots of unity). Previous work using the Tate sequence of E and the Chinburg class Ω m has shown that the stable isomorphism class of E is determined by the data ∆S, µ µ µ, Ω m , and a special character ε of H 2 G, Hom(∆S, µ µ µ) . This paper explains how to build a G-module M from this data which is stably isomorphic to E ⊕ ZG n , for some integer n.Let K/k be a finite Galois extension of number fields with Galois group G and let S be a finite G-stable set of primes of K containing all archimedean primes. Assume that S is large in the sense that it contains all ramified primes of K/k and that the S-class group of K is trivial. Let E denote the G-module of S-units of K and µ µ µ the roots of unity in K. The purpose of this paper is to specify the stable isomorphism class of the G-module E in a much more explicit way than in Theorem B of [7].More precisely, and continuing in the notation of [7], we recall that [11], [12] defines a canonical 2-extension class of G-modules, represented by Tate sequenceswith A a finitely generated cohomologically trivial ZG-module, B a finitely generated projective ZG-module and ∆S the kernel of the G-map ZS → Z which sends every element of S to 1.

show abstract

Exact Sequences and Galois Module Structure

Cited by 64 publications

References 9 publications

On a Multiplicative–Additive Galois Invariant and Wildly Ramified Extensions

On a Multiplicative–Additive Galois Invariant and Wildly Ramified Extensions

Computations in Relative Algebraic K-Groups

Galois structure ofS-units

Contact Info

Product

Resources

About