Special generating sets of purely inseparable extension fields of unbounded exponent (Q1069989)

Let \(K\) be a field of characteristic \(p\neq 0\). This paper seeks to solve an open problem posed by M. Weisfeld in 1965 asking for a necessary and sufficient condition for a purely inseparable extension L/K of unbounded exponent to have a maximal subfield having a subbasis over K. The bounded exponent case had been determined by Weisfeld. A subset M of L is called distinguished in L/K if by arranging M in the form \(M=A_ 1\cup A_ 2\cup \cdots\) where \(A_ i\) consists of the elements of M having exponent i over \(K\) we have (a) M is a subbasis for K(M), (b) the subsets \(M_ i=\cup^{\infty}_{j=i+1}A_ j\), \(i=0,1,\ldots\), satisfy \(M_ i^{p^ i}\) is a minimal generating set for \(K(M^{p^ i}),\) (c) \(K\cap L^{p^ n}\subseteq K^ p(A^ p_ n\cup A^ p_{n+1}\cup \ldots)\), \(n=0,1,\ldots.\) The main result is that the extension L/K has a maximal subfield having a subbasis over \(K\) if and only if L/K has a distinguished subset M.