Galois Groups Over Nonrigid Fields
Wenfeng Gao, David B. Leep, Jan Minac, and Tara L. Smith
Let $F$ be a field with ${\rm char} F \neq 2$. We show that $F$ is a nonrigid field if and only if certain small $2$-groups occur as Galois groups over $F$. These results provide new "automatic realizability" results for Galois groups over $F$. The groups we consider demonstrate the inequality of two particular metabelian $2$-extensions of $F$ which are unequal precisely when $F$ is a nonrigid field. Using known results on connections between rigidity and existence of certain valuations, we obtain Galois-theoretic criteria for the existence of these valuations.