Quadratic Corestriction, -Embedding Problems, and Explicit Construction

Swallow, John; Thiem, F. Nathaniel

doi:10.1081/agb-120004485

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Applications Of Embedding Criteria To Galois Realizability3

Construction Of Cyclic Extensions Of Degree 3 or 9 Without T1

Introduction1

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2004

2011

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Cited by 8 publications

(5 citation statements)

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“…Similarly, conditions for the direct products C 4 × C 8 , C 4 × D 4 , and C 4 × Q 8 are easily derived from the conditions for the realizability of each factor. Conditions for eleven more groups of order 32 can be found in (or easily derived from results in) [2,8,9]. We give results for 13 additional groups of order 32, leaving only 10 without known realizability conditions.…”

Section: Applications Of Embedding Criteria To Galois Realizabilitymentioning

confidence: 97%

Galois realizability of non-split group extensions of C2 by (C2)r×(C4)s×(D4)t

Grundman¹,

Stewart²

2004

Journal of Algebra

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Section: Applications Of Embedding Criteria To Galois Realizabilitymentioning

confidence: 97%

Galois realizability of non-split group extensions of C2 by (C2)r×(C4)s×(D4)t

Grundman¹,

Stewart²

2004

Journal of Algebra

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“…Similarly expressed conditions to the embedding problems given by the towers 5 through 7 may be derived from the results of QUATERNION ALGEBRAS 4539 [26], which gives obstructions for the tops of those towers, and conditions for towers 1 through 4 may be derived from results surveyed in [8]. Precise descriptions, including the explicit determination of the action of the Galois groups on elements of the ®eld extensions, of the Z=2Z-embedding problems occurring at the bottoms of the 14 towers are given in [26].…”

Section: Applications To Solvability Conditions For Embedding Problemsmentioning

confidence: 99%

Quadratic Descent for Quaternion Algebras

Swallow

2001

Communications in Algebra

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Given a Galois embedding problem H 3 G GalL=F with kernel of order two and F Hilbertian, we consider how obstructions O K to subgroup embedding problems H 0 3 G 0 GalL=K for K : F 2 descend to the obstruction O F to the original embedding problem, up to Br 2 K=F. In particular, to such an obstruction O K we associate a tower of Z=2Z-embedding problems and prove that the contribution of O K to O F is given by the obstruction to the last embedding problem in the tower. We show that such an association in fact holds generally for central, Brauer Z=pZ-problems. When O K is the class of a quaternion algebra, we give an explicit representation of O F up to Br 2 K=F. As a consequence, we represent in terms of quaternion algebras over F the obstructions to Z=2Z Â Z=2Z-embedding problems not previously determined over F. 4523

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“…For general primitive elements like the preceding x 1 of Lemma 7.4, see [15] or [20]. For a kernel of order 2, see also [21]. …”

Section: Construction Of Cyclic Extensions Of Degree 3 or 9 Without Tmentioning

confidence: 99%