Extending Automorphisms and Derivations onto Ore-Extensions

Abstract

We study the question wether an automorphism σ of a field K can be extended to an automorphism τ of the field of fractions ${Q=K(x; \alpha, \delta)}$ of the Ore-extension ${R=K[x; \alpha, \delta]}$ (Sect. 3) and wether a σ-derivation ${\varepsilon}$ of K can be extended to a τ-derivation of Q (Sect. 4), and determine all extensions of σ and ${\varepsilon}$. Until now these question have only been discussed under special assumptions (for example in [7] and [11]). In particular, little is known on extensions of derivations. The result is in each case a criterion for extendability (Lemmata 3.2 and 4.3). The characterization of all extensions of automorphisms σ from K to Q is well understood (Corollary 3.5 and Theorem 3.12). This is in contrast to the characterization of the extensions of σ-derivations ${\varepsilon}$, which can only be described satisfactorily under additional assumptions (Theorems 4.8, 4.9, 4.10). We obtain the set of all extensions of σ or ${\varepsilon}$ easily from a particular extension and the normalizer ${N(\sigma^{-1} \alpha \, \sigma)}$ or ${N(\alpha \, \sigma)}$ (Corollaries 3.3 and 4.4). These normalizers will be described in Sect. 2 by minimal elements of R.