This talk will be a report on joint work with Gabi Nebe, begun during her visit to Sydney in March.

We deal with characteristic zero representations and characters of a finite group G. The Schur index of an absolutely irreducible character is the minimal degree of an extension of the character field so that there is a representation of G over the extension field that affords the character. The Schur index also lets us work out the characters of irreducible representations over a given field. Both these issues are important and can be difficult to resolve when computing characteristic zero representations.

We aim for an algorithm that computes Schur indices using character theory and computation in G, i.e. without computing any representations, for use in resolving the issues above.

The problem has a long history and there are plenty of published results that we can use. In this talk I will give a brief and biased survey of the theory and discuss how we have put together such an algorithm for computing Schur indices over subfields of cyclotomic fields and their completions.