Non-Abelian Groups of Order Eight and the Local LiftingProblem

Abstract

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this thesis, we characterize D_4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D_4-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D_4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D_4 is a local Oort group for the prime 2. Furthermore, we characterize Q_8-extensions of fields of characteristic two, determine the ramification breaks of (suitable) Q_8-extensions of complete discrete valuation fields of characteristic two, and, by solving the local lifting problem in the negative for a family of Q_8-extensions of complete discrete valuation fields of characteristic two with algebraically closed residue field, show that neither Q_8 nor SL_2(Z/3Z) is an almost local Oort group for the prime 2.