+ v : RngValElt -> RngValElt
- v : RngValElt -> RngValElt
v + w : RngValElt, RngValElt -> RngValElt
v - w : RngValElt, RngValElt -> RngValElt
v * w : RngValElt, RngValElt -> RngValElt
v ^ k : RngValElt, RngIntElt -> RngValElt
v / w : RngValElt, RngValElt -> RngValElt
v +:= w : RngValElt, RngValElt -> RngValElt
v -:= w : RngValElt, RngValElt -> RngValElt
v *:= w : RngValElt, RngValElt -> RngValElt
The quotient q of the division with remainder v=qw + r of the valuation
ring elements v and w, where the
remainder will have valuation less than that of w; if the valuation of
v is greater than or equal than that of w, this simply returns
the quotient v/w, if the valuation of w exceeds that of v it
returns 0.
v eq w : RngValElt, RngValElt -> BoolElt
v ne w : RngValElt, RngValElt -> BoolElt
v in V : RngValElt, Rng -> BoolElt
v notin V : RngValElt, Rng -> BoolElt
Parent(v) : RngValElt -> RngVal
Category(v) : RngValElt -> Cat
IsZero(n) : RngValElt -> BoolElt
IsOne(n) : RngValElt -> BoolElt
IsMinusOne(n) : RngValElt -> BoolElt
IsNilpotent(n) : RngValElt -> BoolElt
IsIdempotent(n) : RngValElt -> BoolElt
IsUnit(n) : RngValElt -> BoolElt
IsZeroDivisor(n) : RngValElt -> BoolElt
IsRegular(n) : RngVal -> BoolElt
Valuation(v) : RngValElt -> RngIntElt
Given an element v of a valuation ring V, return the valuation
(associated with V) of v.
Given two elements v, w of a valuation ring V with associated
valuation φ, return a quotient and remainder q and r in
V such that v=qw + r and 0≤φ(r)<φ(w). If φ(v)<φ(w)
this simply returns q=0 and r=v, and if φ(v)≥φ(w) then
it returns q=v/w and r=0.
Gcd(v, w) : RngValElt, RngValElt -> RngValElt
GCD(v, w) : RngValElt, RngValElt -> RngValElt
This function returns a greatest common divisor of two elements
v, w in a valuation ring V. This will return um, where
m=min(φ(v), φ(w)) is the minimum of the valuations
of v and w m=min(φ(v), φ(w)) and u is the uniformizing
element of V (with valuation φ(u)=1).
Xgcd(v, w) : RngValElt, RngValElt -> RngValElt, RngValElt, RngValElt
XGCD(v, w) : RngValElt, RngValElt -> RngValElt, RngValElt, RngValElt
This function returns a greatest common divisor z∈V of two elements
v, w in a valuation ring V as well as multipliers x, y∈V
such that xv + yw=z. The principal return value will be z=um, where
m=min(φ(v), φ(w)) is the minimum of the valuations
of v and w m=min(φ(v), φ(w)) and u is the uniformizing
element of V (with valuation φ(u)=1).
V2.28, 13 July 2023