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