On centralizer-related intermediate fields of a skew field extension
Let \(L/K\) be a skew field extension, \(D\) an intermediate field of \(L/K\), not necessarily commutative, and \(Z_L(D)\) the centralizer of \(D\) in \(L\). The paper under review singles out four elementary types of intermediate fields of \(L/K\) with respect to set-theoretic inclusion. They are determined by the four extreme possibilities arising from the inclusions \(K\subseteq K\cdot Z_L(K)\subseteq L\) and \(K\subseteq Z_LZ_L(K)\subseteq L\). As noted by the author, the equality \(Z_LZ_L(K)=L\neq K\) is impossible when \(L\) is of finite dimension over its centre \(Z(L)\) (by the double centralizer theorem). A considerable part of the main results of the paper are obtained under the assumption that \(L\) is finite-dimensional as a left vector space over \(K\). Besides providing characterizations of the basic types, the paper shows that \(L/K\) decomposes into four extensions of these types. More precisely, it turns out that there are two forms of such a decomposition, and it becomes clear that they are dual to each other. In addition, it is proved that the presentation of \(L/K\) in each of these two forms is unique.
scientific article; zbMATH DE number 1750362