Galois Groups Over Nonrigid Fields
Wenfeng Gao, David B. Leep, Jan Minac, Tara L. Smith
Abstract
Let F be a field with characteristic ≠ 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.
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