Composites of algebraically closed fields
Let \(k\) be an algebraically closed field, and suppose that \(K\) is a composite of extension fields \(K_i\) which are linearly disjoint over \(k\). The author proves that if \(k\subseteq L\subseteq K\), where \(L\) is a field with \(L^q=L\) for some prime \(q\) not equal to the characteristic of the fields, then \(L\subseteq K_i\) for some \(i\). The proof is obtained from the study of composites of two linearly disjoint, algebraically closed extensions of \(k\); and some interesting properties of such a composite are demonstrated. In particular, purity of an element \(z\) of a composite \(K_1K_2\), i.e. when \(z=z_1z_2\) for \(z_ i\in K_i\) is investigated.
linearly disjoint, algebraically closed extensions
scientific article; zbMATH DE number 4156595