Subsections

• Creation of Structures
• Creation of Elements

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) >= 0, that is, r=(x/y) in Q such that p not| y.

ValuationRing(F, f) : FldFun, 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 of rational functions (g/h) in F with v_f(g/h) >= 0, that is, with f not| h.

ValuationRing(F) : FldFun -> 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_(Infinity), consisting of (g/h) in F such that deg(h) >= deg(g).

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.