Sentence examples for discrete valuation from inspiring English sources

Let (mathcal K ) be a field complete with respect to a discrete valuation (v) with residue field (k).

Let (K/F) be a finite separable field extension, (v) a discrete valuation of (F), and (w) an extension of (v) to (K).

Given a discrete valuation (v) of (K), we will denote by (mathcal O _{K,v}) and (overline{K}_v) its valuation ring and residue field, respectively.

We are given a complete discrete valuation field K of valuation ring O = O K and of perfect residue field k = O / π O.

We recall that given a finite-dimensional central division algebra (mathcal D ) over a field (mathcal K ) which is complete with respect to a discrete valuation (v), the valuation (v) uniquely extends to a discrete valuation (tilde{v}) of (mathcal D ) (cf. [27, Chap. 12, §2], [33]).

Given a discrete valuation (v) of (K) and a maximal (K_v -torus (T_v) of (G), there exists a maximal (K_v -torusT_vof (G) which is generic over (K) and is conjugathere (T_v) by an existst of (G(K_v)).

Let (K) be a field, and (V) be a set of discrete valuations of (K) satisfying conditions (A), (B) and (C) for a given integer (n > 1).

We now introduce the following set of discrete valuations of (K): begin{aligned} V = V_0 cup V_1, end{aligned}where (V_0) is the set of all geometric places of (K) (i.e., those discrete valuations that are trivial on (k)), and (V_1) consists of the valuations (w v)) for all (v in V^k{setminus }S).

Indeed, according to Theorem 8 in [5], there exists a set (V) of discrete valuations of (K) that satisfies conditions (A), (B) and (C) and for which the unramified Brauer group (_nmathrm{Br}(K _V) is finite.

In this paper, we describe a general approach to proving the finiteness of (mathbf{gen}(D)) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of (K).

The proof consists of two parts: first, one relates the size of (mathbf{gen}(D)) to that of (_nmathrm{Br}(K _V), the (n -torsion of the un -torsionBrauer grofp (mathem{Br}(K)_V) with respect to a sunramifiedt (V) of discrete valuations of (K); second, one estaBrauers the finiteness of (_nmathrm{Br}(K)_V).