GALOIS GROUPS OVER NONRIGID FIELDS

WENFENG GAO DAVID B. LEEP JAN MINAC TARA L. SMITH

Abstract. Let F be a field with char F ≠ 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 Galoistheoretic criteria for the existence of these valuations.