Nonexcellent finite field extensions of 2-primary degree

Abstract

Let \(k_0\) be a field of characteristic distinct from 2, \(l_0{/}k_0\) a finite field extension of degree \(2^m\), \(m\ge 2\). We prove that there exists a field extension \(K{/}k_0\) linearly disjoint with \(l_0/k_0\) such that the extension \(l_0K{/}K\) is nonexcellent. The crucial point is a construction of in some sense nonstandard elements in \(H^3(K,\mathbb {Z}{/}2\mathbb {Z})\). We apply this construction as well for investigation of the group \({F^*}^2N_{L{/}F}L^*\), where L / F is a \(\mathbb {Z}{/}4\mathbb {Z}\times \mathbb {Z}{/}2\mathbb {Z}\)-Galois extension. More precisely, let \(F\subset F_i\subset L\)\((1\le i\le 3)\) be the three intermediate quadratic extensions of F, and \(N_i(F)=N_{F_i{/}F}F_i^*\). We show that the quotient group \({N_1(F)\cap N_2(F)\cap N_3(F)\over {F^*}^2N_{L/F}L^*}\) can be arbitrarily large.