Characterizing rosy theories

We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

The Maximal p-Extension of a Local Field

1. Let k denote a local field, that is, a complete discrete-valued field with perfect residue class field . Let G denote the Galois group of the maximal separable algebraic extension M of k, and let g denote the corresponding object over . For a given prime integer p, let G(p) denote the Galois group of the maximal p-extension of k. The dimensions of the cohomology groupsconsidered as vector spaces over the prime field Z/pZ, are equal, respectively, to the rank and the relation rank of the pro-p-group G(p); see [4; 9]. These dimensions are well known in many cases, especially when k is finite [6; 3; (Hoechsmann) 2, pp. 297-304], but also when k has characteristic p, or when k contains a primitive pth root of unity [4, p. 205].