Element Operations

Arithmetic Operations

+ 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

v div 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 \(0\).

Equality and Membership

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 and Category

Parent(v): RngValElt -> RngVal

Category(v): RngValElt -> Cat

Predicates on Ring Elements

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

Other Element Functions

EuclideanNorm(v): RngValElt -> RngIntElt

Valuation(v): RngValElt -> RngIntElt

Given an element \(v\) of a valuation ring \(V\), return the valuation (associated with \(V\)) of \(v\).

Quotrem(v, w): RngValElt, RngValElt -> RngValElt, RngValElt

Given two elements \(v, w\) of a valuation ring \(V\) with associated valuation \(\phi\), return a quotient and remainder \(q\) and \(r\) in \(V\) such that \(v=qw+r\) and \(0\leq\phi(r)<\phi(w)\). If \(\phi(v)<\phi(w)\) this simply returns \(q=0\) and \(r=v\), and if \(\phi(v)\geq\phi(w)\) then it returns \(q=v/w\) and \(r=0\).

GreatestCommonDivisor(v, w): RngValElt, RngValElt -> RngValElt

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 \(u^m\), where \(m=\min(\phi(v), \phi(w))\) is the minimum of the valuations of \(v\) and \(w\) and \(u\) is the uniformizing element of \(V\) (with valuation \(\phi(u)=1\)).

ExtendedGreatestCommonDivisor(v, w): RngValElt, RngValElt -> RngValElt, RngValElt, RngValElt

Xgcd(v, w): RngValElt, RngValElt -> RngValElt, RngValElt, RngValElt

XGCD(v, w): RngValElt, RngValElt -> RngValElt, RngValElt, RngValElt

This function returns a greatest common divisor \(z\in V\) of two elements \(v, w\) in a valuation ring \(V\) as well as multipliers \(x, y\in V\) such that \(xv+yw=z\). The principal return value will be \(z=u^m\), where \(m=\min(\phi(v), \phi(w))\) is the minimum of the valuations of \(v\) and \(w\) and \(u\) is the uniformizing element of \(V\) (with valuation \(\phi(u)=1\)).