Algorithms for function fields

Abstract

Let {\ASIE K}\,/{\small $\Q$}({\ASIE t \!}) be a finite extension. We describe algorithms for computing subfields and automorphisms of {\ASIE K}\,/{\small $\Q$}({\ASIE t }\!). As an application we give an algorithm for finding decompositions of rational functions in {\small $\Q(\alpha)$}. We also present an algorithm which decides if an extension {\ASIE L}\,/{\small $\Q$}({\ASIE t \!}) is a subfield of {\ASIE K}. In case [{\ASIE K : \;}{\small$\Q$}({\ASIE t \!})] = [{\ASIE L : \;}{\small $\Q$}({\ASIE t \!})] we obtain a {\small $\Q$}({\ASIE t \!})-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of {\ASIE K}\,/{\small $\Q$}({\ASIE t \!}).