Given the rational field Q and a rational prime number p, create
the valuation ring R corresponding to the discrete non-Archimedean
valuation vp, consisting of rational numbers r such
that vp(r)≥0, that is, r=(x/y)∈Q such that p not| y.
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∈K[x], create
the valuation ring R corresponding to the discrete non-Archimedean
valuation vf. Thus R consists of rational functions (g/h)∈F
with vf(g/h)≥0, that is, with f not| h.
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∞, consisting of
(g/h)∈F such that deg(h)≥deg(g).
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.
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∈F for which the
valuation on V is non-negative, an error occurs if this is not the case.
V2.28, 13 July 2023