On the image of derivations

Abstract

Introduction. Let L be a field. An additive mapping d : L −→ L is said to be a derivation of L if d(ab) = d(a)b+ ad(b) for any a, b ∈ L. If K ⊂ L is a field extension, then a derivation d of L is said to be a K-derivation if d(αa) = αd(a) for any α ∈ K and a ∈ L. If d is a derivation of L such that d(L) = L, then we say that d is epimorphic. (a) Let K ⊂ L be a finite dimensional field extension and let d be a Kderivation of L. Then L has a direct sum decomposition L = Ker d ⊕ L′, as a K-module, where L′ is K-isomorphic to d(L). Thus dimK L > dimK d(L) yields d(L) 6= L for any K-derivation d. (b) Assume now that L is a purely transcendental extension with infinite transcendental basis {xλ; λ ∈ Λ} over Q, the field of rational numbers. Since |L| = |Λ| (the cardinalities of L and Λ), we may put L = {aλ; λ ∈ Λ}. Then the Q-derivation d of L defined by d(xλ) = aλ is epimorphic. Thus it is natural to ask whether a field extension K ⊂ L has (or has not) an epimorphic K-derivation. The purpose of this paper is to study on this problem.