Creation Functions

Creation of Structures

ValuationRing(Q, p): FldRat, RngIntElt -> RngVal

Given the rational field \(Q\) and a rational prime number \(p\), create the valuation ring \(R\) corresponding to the discrete non-Archimedean valuation \(v_p\), consisting of rational numbers \(r\) such that \(v_p(r)\geq 0\), that is, \(r={x\over y}\in{\mathbb{Q}}\) such that \(p\not\vert\, y\).

ValuationRing(F, f): FldFunRat, RngUPolElt -> RngVal

Given the rational function field \(F\) as a field of fractions of the univariate polynomial ring \(K[x]\) over a field \(K\), as well as a monic irreducible polynomial \(f\in K[x]\), create the valuation ring \(R\) corresponding to the discrete non-Archimedean valuation \(v_f\). Thus \(R\) consists of rational functions \({g\over h}\in F\) with \(v_f(g/h)\geq 0\), that is, with \(f\not\vert\, h\).

ValuationRing(F): FldFunRat -> RngVal

Given the rational function field \(F\) as a field of fractions of the univariate polynomial ring \(K[x]\) over a field \(K\), create the valuation ring \(R\) corresponding to \(v_{\infty}\), consisting of \({g\over h}\in F\) such that \(\deg(h)\geq\deg(g)\).

ValuationRing(K, p): FldNum, RngOrdIdl -> RngVal

ValuationRing(K, p): FldFun, RngFunOrdIdl -> RngVal

Given an algebraic number field or function field \(K\) and a prime ideal \(p\) contained in \(K\), construct the valuation ring \(R\) corresponding to the valuation given by the prime ideal \(p\).

Creation of Elements

V ! r: RngVal, FldFunElt -> RngValElt

V ! r: RngVal, FldRatElt -> RngValElt

Given a valuation ring \(V\) and an element of the field of fractions \(F\) of \(V\) (from which \(V\) was created), coerce the element \(r\) into \(V\). This is only possible for elements \(r\in F\) for which the valuation on \(V\) is non-negative, an error occurs if this is not the case.