Hilbert 90 theorems over division rings

Abstract:

Hilbert’s Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobson’s criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K, S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions $K[t,S,D]$, we characterize the notion of (S, D)-algebraicity for elements $a \in K$, and give an effective criterion for two elements $a,b \in K$ to be (S, D)-conjugate, in the case when the (S, D)-conjugacy class of a is algebraic. This criterion amounts to a general Hilbert 90 Theorem for division rings in the (K, S, D)-setting, subsuming and extending all known forms of Hilbert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.