TY - JOUR

T1 - Extending Automorphisms and Derivations onto Ore-Extensions

AU - Karpfinger, Christian

AU - Koehler, Henning

AU - Wähling, Heinz

N1 - Publisher Copyright:
© 2015, Springer Basel.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - We study the question wether an automorphism σ of a field K can be extended to an automorphism τ of the field of fractions (Formula Presented.) of the Ore-extension (Formula Presented.) (Sect. 3) and wether a σ-derivation (Formula Presented.) of K can be extended to a τ-derivation of Q (Sect. 4), and determine all extensions of σ and (Formula Presented.). 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 (Formula Presented.), 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 (Formula Presented.) easily from a particular extension and the normalizer (Formula Presented.) or (Formula Presented.) (Corollaries 3.3 and 4.4). These normalizers will be described in Sect. 2 by minimal elements of R.

AB - We study the question wether an automorphism σ of a field K can be extended to an automorphism τ of the field of fractions (Formula Presented.) of the Ore-extension (Formula Presented.) (Sect. 3) and wether a σ-derivation (Formula Presented.) of K can be extended to a τ-derivation of Q (Sect. 4), and determine all extensions of σ and (Formula Presented.). 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 (Formula Presented.), 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 (Formula Presented.) easily from a particular extension and the normalizer (Formula Presented.) or (Formula Presented.) (Corollaries 3.3 and 4.4). These normalizers will be described in Sect. 2 by minimal elements of R.

KW - Derivation

KW - Ore-extension

KW - Skew field

KW - Skew polynomial ring

UR - http://www.scopus.com/inward/record.url?scp=84944173232&partnerID=8YFLogxK

U2 - 10.1007/s00025-015-0447-1

DO - 10.1007/s00025-015-0447-1

M3 - Article

AN - SCOPUS:84944173232

SN - 1422-6383

VL - 68

SP - 395

EP - 413

JO - Results in Mathematics

JF - Results in Mathematics

IS - 3-4

ER -