Intersections of two cofinite subfields
Let \(A\) and \(B\) be subfields of a field \(F\). Let \(P\) denote the prime subfield of \(F\). Then it is known that it is possible for \(F\) to be infinite dimensional over \(A\cap B\) even if \(F/A\) and \(F/B\) are finite-dimensional. In this paper, the author shows that if \(F\) is finitely generated over \(P\), purely inseparable over \(A\) or \(B\), and finite-dimensional over both \(A\) and \(B\), then \(F\) is finite dimensional over \(A\cap B\). The author shows that \(F\) may be infinite dimensional over \(A\cap B\) if the assumption that \(F\) is finitely generated over \(P\) is removed. In fact, the author shows that if \(P\) is of characteristic \(p\) and \(F\) is generated over \(P\) by indeterminates \(x_1,x_2,\dots, y_1,y_2,\dots\), then there are subfields \(A\), \(B\), \(C\) with \(F\) Galois of degree 2 over \(A\) and purely inseparable of degree \(p\) over \(B\) and over \(C\), where \(F\) has infinite dimension over \(A\cap B\) and \(B\cap C\).
purely inseparable finitely generated field extension
scientific article; zbMATH DE number 2106403