An extension of Dedekind’s linear independence theorem

Abstract

Dedekind's theorem on the linear independence of isomorphisms of a field is extended to the case of linear independence of compositions of isomorphisms and powers of a derivation, D,, for fields of characteristic zero which contain an element s such that D(s)=1. In this paper we present an extension of the following classical theorem on the linear independence of isomorphisms of a field [1, p. 25]. THEOREM 1 [DEDEKIND]. Let E and P be fields. Then the collection {FJF is an isomorphism of E into P} is left linearly independent over P; i.e., 1pip F=O, pi in P, n> 1 an integer, and Fi 0 Fj for i #j implies pi=O for i= I , 2, ***,n. For fields of characteristic zero, the above theorem may be extended to the case of compositions of isomorphisms and powers of a derivation on E as in the following theorem. This result is useful in the study of linear transformations on the field of Mikusinski operators [2]. THEOREM 2. Let E and P befields of characteristic zero, D a derivation of E into E, and s an element in E such that D(s)= 1. Then, the collection {FD'IF is an isomorphism of E into P and n>O is an integer} is left linearly independent over P. In the following we consider elements of a field E as endomorphisms of E defined for b in E by b(x)=bx for each x in E. (A field E can be embedded in the ring of group endomorphisms on the additive group E.) For the proof of Theorem 2, we will need a lemma. LEMMA. Let E and P be fields of characteristic zero, D a derivation of E into E, and s an element in E such that D(s)= 1. If F and G are homomorphisms of E into P such that FDv=GD7 for some integers p, q>O, then p=q and F=G. Received by the editors August 28, 1972. AMS (MOS) subject classqifcations (1970). Primary 13B10.