Rational twisted series

Abstract

Rational twisted power series over a (commutative) filed are studied. We give several characterizations of such series which are similar to the classical results concerning rational power series over a commutative field. In particular, we prove a version of Kronecker's lemma for the rationality of twisted power series.

Algebraically closed $σ$-fields

“…be an anti-automorphism. First, we prove (1). Since the set of units of K[T ; σ] is K * , we see that α(K) ⊂ K. Similarly, we have α −1 (K) ⊂ K since α −1 is an anti-automorphism of K[T ; α].…”

“…In the case when σ is an involution, the identity map of K[T ; σ] belongs to AAut(K, σ). In this case, an application of this proposition gives the following: (1)…”

“…To prove the second statement, let α 0 : K → K be an automorphism satisfying (1) and (3). Given a 0 ∈ K * , we define α :…”

“…We note that (K α , σ) is a σ-extension of (K β , σ) if β ≤ α ≤ α 0 . Now, we consider the σ-field (K (1) , σ) = (K α 0 , σ). Clearly, (K (1) , σ) is a σ-extension of (K, σ).…”

“…Now, we consider the σ-field (K (1) , σ) = (K α 0 , σ). Clearly, (K (1) , σ) is a σ-extension of (K, σ). By construction, every P (T ) ∈ D has a root in K (1) .…”