Timo Hanke

A twisted Laurent series ring that is a noncrossed product

The striking results on noncrossed products were their existence (Amitsur [1]) and the determination of ℚ(t) and ℚ((t)) as their smallest possible centres (Brussel [3]). This paper gives the first fully explicit noncrossed product example over ℚ((t)). As a consequence, the use of deep number theoretic theorems (local-global principles such as the Hasse norm theorem and density theorems) in order to prove existence is eliminated. Instead, the example can be verified by direct calculations. The noncrossed product proof is short and elementary.

Galois subfields of inertially split division algebras

Abstract Let D be a valued division algebra, finite-dimensional over its center F. Assume D has an unramified splitting field. The paper shows that if D contains a maximal subfield which is Galois over F (i.e. D is a crossed product) then the residue division algebra D ¯ contains a maximal subfield which is Galois over the residue field F ¯ . This theorem captures an essential argument of previously known noncrossed product proofs in the more general language of noncommutative valuations. The result is particularly useful in connection with explicit constructions.

On absolute Galois splitting fields of central simple algebras

Abstract A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p -algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald–Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.

Complexity of Elements in Rings

If R is a ring with subset S then the rational closure DivR(S) of S in R is the smallest subring D of R containing S such that U(D)=D∩U(R) where U(D), resp. U(R), denotes the group of units of D resp. R. In this paper a new approach to the so-called complexity is given in order to describe how the elements of DivR(S) are built from elements of S.

The isomorphism problem for cyclic algebras and an application

The isomorphism problem means to decide if two given finite-dimensional simple algebras with center K are K-isomorphic and, if so, to construct a K-isomorphism between them. Applications lie in computational aspects of representation theory, algebraic geometry and Brauer group theory. The paper presents an algorithm for cyclic algebras that reduces the isomorphism problem to field theory and thus provides a solution if certain field theoretic problems including norm equations can be solved (this is satisfied over number fields). As an application, we can compute all automorphisms of any given cyclic algebra over a number field. A detailed example is provided which leads to the construction of an explicit noncrossed product division algebra.