by Wieland Fischer (doctoral thesis, July 1999, 143 pages)
Twisted cyclic algebras are triples (A,L,K) where A is a central simple algebra and L, K are maximal commutative subalgebras of A with the following property: Over the algebraic closure the triple (A,L,K) becomes isomorphic to a standard triple consisting of a matrix algebra, the diagonal subalgebra, and the centralizer of a permutation matrix of order n=deg A.
The automorphism group of this split triple is a semi-direct product (un×Z/n).Aut(Z/n), where un is the group of n-th roots of unity. Twisted cyclic algebras form a natural framework for the chain equivalence of Kummer elements (cf. The chain lemma for Kummer elements of degree 3.)
Various varieties related with Kummer elements are constructed. It is shown that for algebras of degree 3 the chain lemma of length 3 for Kummer elements holds over a quadratically closed field. More precisely, one proves that for two generic Kummer elements there exist exactly two chains connecting them, where these chains may be possibly defined only over a quadratic extension. This complements the results of loc. cit., where the chain lemma of length 4 is proven.
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