HILBERT-90 THEOREMS OVER DIVISION RINGS

Hilbert's Satz 90 is well-known for cyclic extensions of fields, but a ttempts at generalizations to the case of division rings have only bee n partly successful. Jacobson's criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 9 0, 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)-al gebraicity for elements a is-an-element-of K, and give an effective cr iterion for two elements a, b is-an-element-of K to be (S, D)-conjugat e, 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 H ilbert 90 in the literature, including the aforementioned Jacobson Cri terion. Two of the working tools used in the paper, the Conjugation Th eorem (2.2) and the Composite Function Theorem (2.3), are of independe nt interest in the theory of Ore extensions.