The explicit construction of noncrossed product algebras over small fields

The existence of so-called "noncrossed products" was a major open question about finite-dimensional division algebras before it was answered by Amitsur in 1972. A division algebra D, finite-dimensional over its centre Z(D), is called a "crossed product" if it contains a maximal commutative subfield that is a Galois extension of the field Z(D), and otherwise a "noncrossed product". Crossed product algebras possess a basis whose multiplication table can be described in terms of a Galois action and a 2-cocycle. Noncrossed products lack such a description.

The smallest centres over which noncrossed products arise are the rational function field Q(t) and the Laurent series field Q((t)). This was discovered by E. Brussel in 1995. The talk discusses the difficulties that are involved in constructing explicit examples over these fields, presents a new approach that is algorithmically accessible, and gives an explicit example of degree 9 that was computed with the help of a computer algebra system. Here, "explicit" means that structure constants (a multiplication table) of the noncrossed product is known.