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.