Submission: 2002, Jul, 26
We discuss cohomology groups $H^1(F,\mu_n)$ with various actions of $Gal(F_s/F)$ on $\mu_n$, and relate them to eigenspaces of the group $H^1(K,\mu_n)$ where $K = F[\rho_n]$, and to cyclic field extensions of $K$ which are Galois over $F$. We assume that $[K:F]$ is prime to $n$. Various applications to the second cohomology groups and to central simple algebras are given. The results include extensions of Albert's cyclicity criterion for division algebras of prime degree, to algebras of prime-power degree over $F$. We also extend the Rosset-Tate result on the corestriction of cyclic algebras in the presence of roots of unity, to extensions in which roots of unity live in an extension of dimension $\leq 3$ over the base field. In particular if roots of unity live in a quadratic extension of the base field, then corestriction of a cyclic algebra along a quadratic extension is similar to a product of two cyclic algebras. Moreover, dihedral algberas over such field are cyclic. We also construct generic examples of algebras which become cyclic after extending scalars by roots of unity, and show the existence of elements for which most powers have reduced trace zero.
2000 Mathematics Subject Classification: 16K20, 12G05
Keywords and Phrases: Galois cohomology, Albert cyclicity criterion, semidirect crossed product, quasi-symbols
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