In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point on the curve. More examples can be found in the article on discrete valuation rings . every complete discrete valuation ring o with finite residue field of cardinality q>nand odd characteristic, the degrees of the regular characters of G(o)of level are q(n 2)(−1) v ε G (q)/u τ ε (q),τ∈A n. In particular, the degrees of the regular … over commutative local rings, speci cally mentioning discrete valuation rings. Valuation Rings111 K there is a minimum exponent of t occurring in f, which implies that we can write f uniquely as f =tn g for some n 2Z and g 2L[[t]] with non-zero constant coefficient. In particular, this holds for regular quadratic forms over a local ring Rand for regular symmetric bilinear forms over a local ring … 12. regular local rings all having dimension equal to the dimension of R[A, Lemma 12]. Any complete local ring can be represented as the quotient ring of the ring $ S [ [ X _ {1} \dots X _ {n} ] ] $ of formal power series, where $ S $ is a field (in the case of equal characteristic) or of a complete discrete valuation ring … In this article we prove that certain rank-one discrete valuation rings (DVRs) can be represented as a directed union of regular … phic image of a formal power series ring over a complete discrete valuation ring (V;pV) whose maximal ideal is generated by a positive prime integer p. If a mixed characteristic local ring is a domain, it is module- nite over a formal power series ring over such a ring … With this definition K is in fact a field: it is obvious that it is a ring… A discrete valuation ring (DVR) is an integral domain that is the valuation ring of its fraction eld with respect to a discrete valuation. It is easy to verify that every valuation ring Ais a in fact a ring, and even …