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 \(\phi(u)=1\)).