Algorithms for function fields

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(α)}. 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 \!}).