Valued fields with contractive automorphism and Kaplansky fields
The author determines the elementary theories of valued fields equipped with a contractive automorphism. This is achieved under certain technical assumption on the residue difference field. Precisely, he proves the following: Theorem: Let \(K\) and \(K'\) be \(\sigma\)-Henselian valued difference fields with contractive distinguished automorphisms. Suppose \(K\) and \(K'\) have residue difference fields \(k\) and \(k'\), of characteristic \(0\), and value difference groups \(\Gamma\) and \(\Gamma'\) respectively. Then \(K\equiv K'\) if and only if \(k\equiv k'\), as difference fields, and \(\Gamma\equiv\Gamma'\), as ordered difference groups. The techniques involved can also be applied to valued fields of positive characteristic to obtain alternative proofs of well-known results about Kaplansky fields. For example, the following: Theorem: If \(K\) is a Kaplansky field, then \(K\) has, up to isomorphism over \(K\), a unique algebraically maximal, immediate, algebraic extension.
scientific article; zbMATH DE number 5840419