Relative negligibility of linear automorphisms

Let U and V be finite-dimensional vector spaces over a field k, α∈GL(U), β∈GL(V) and I be the identity transformation on V. Denote by α*β and α*I the induced linear automorphisms on U⊕V; α*β and α*I can also be regarded as k-automorphisms on the function field k(U⊕V). It is elementary to check whether α*β and α*I are conjugate within GL(U⊕V) by examining their rational canonical forms. In this paper we shall give necessary and sufficient conditions for α*β and α*I to be conjugate within Autk(k(U⊕V)). For this characterization, we introduce the concept of the generalized order. Through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field.