A group valuation is constructed on the norm factor group of a quadratic extension of a linearly compact field, and the norm factor group is explicitly computed as a valued group. Generalizations and applications of this structure theory are made to cyclic extensions of prime degree, to square (and pth power) factor groups, to generalized quaternion algebras, and to quadratic extensions of arbitrary fields. Introduction. Let F be a valued field, with valuation v and valuation ring Rv, such that the additive group of F is topologically complete for every (not necessarily Hausdorff) group topology admitting a family of ideals of R, as a neighborhood base at 0. This condition on a valued field is equivalent to Lefschetz's linear compactness (as an R,-module) [13], Krull's maximality [12], and Fleischer's ultracompleteness [6]. These concepts were introduced to generalize the classical local fields. For facts about linearly compact fields (and further equivalent definitions) see [2], [11], [15], [18], [19]. Assuming that F does not have characteristic two and that the residue class field k, of v is perfect if it has characteristic two, we show the field valuation v naturally induces a valuation on the norm factor group of any quadratic extension of F (see ?1 for valuations on groups). We then compute the structure of these factor groups as valued groups. We generalize and apply this computation to the square factor group of F, to generalized quaternion division algebras over F (and in particular their norm factor groups), to quadratic extensions of not necessarily linearly compact fields, and to the norm factor groups of arbitrary cyclic extensions of prime degree. Presented to the Society, January 22, 1971; received by the editors June 1, 1970. AMS 1969 subject classifications. Primary 1398, 1067, 1270; Secondary 1069, 1570, 1245, 1646.