# Elements Operations

## `L ! r: RngLocA, Any -> RngLocAElt`

Return the element of the local field $L$ described by $r$ where $r$ may be anything which is coercible into the quotient representation of $L$.

## `L . i: RngLocA, RngIntElt -> RngLocAElt`

Return the generator of the local field $L$. The only valid input for $i$ is $1$.

## `InertialElement(L): RngLocA -> RngLocAElt`

Return a generator for the inertial subfield of the local field $L$.

## `UniformizingElement(L): RngLocA -> RngLocAElt`

Return an element of the local field $L$ of valuation $1$.

## Arithmetic

### `a * b: RngLocAElt, RngLocAElt -> RngLocAElt`

### `a + b: RngLocAElt, RngLocAElt -> RngLocAElt`

### `a - b: RngLocAElt, RngLocAElt -> RngLocAElt`

### `- a: RngLocAElt -> RngLocAElt`

### `a ^ n: RngLocAElt, RngIntElt -> RngLocAElt`

### `a / b: RngLocAElt, RngLocAElt -> RngLocAElt`

## Predicates on Elements

### `a eq b: RngLocAElt, RngLocAElt -> BoolElt`

Return whether the local field elements $a$ and $b$ are considered equal.

### `IsOne(a): RngLocAElt -> BoolElt`

### `IsMinusOne(a): RngLocAElt -> BoolElt`

Return whether the local field element $a$ is known to be $1$ or $-1$ to the precision of the field.

### `IsWeaklyZero(a): RngLocAElt -> BoolElt`

Return whether the local field element $a$ is not known to be non zero.

### `IsZero(a): RngLocAElt -> BoolElt`

Return whether the local field element $a$ is known to be zero.

## Other Operations on Elements

### `Valuation(a): RngLocAElt -> RngExtReElt`

The valuation of the element $a$ in a local field.

### `RelativePrecision(a): RngLocAElt -> RngExtReElt`

The relative precision of the element $a$ in a local field.

### `Eltseq(a): RngLocAElt -> SeqEnum`

Return the coefficients of powers of the generator of the parent of $a$ in $a$.

### `RepresentationMatrix(a): RngLocAElt -> AlgMatElt`

The representation matrix of the element $a$ of a local field.

### `Norm(a): RngLocAElt -> RngElt`

### `Norm(a, F): RngLocAElt, Rng -> RngElt`

### `Trace(a): RngLocAElt -> RngElt`

### `Trace(a, F): RngLocAElt, Rng -> RngElt`

### `MinimalPolynomial(a): RngLocAElt -> RngUPolElt`

### `MinimalPolynomial(a, F): RngLocAElt -> RngUPolElt`

Return the norm, trace or minimal polynomial of $a$. If a coefficient field $F$ of the parent $L$ of $a$ is given then the norm, trace or minimal polynomial will be that of $a$ as an element of $L$ represented as an extension of $F$.

### `Example: elts (ex-0bf192)`

Continuing from the first example we have :

```magma
> UniformizingElement(L);
a^2 + (6 + O(7^50))*a + 3 + O(7^50)
> InertialElement(L);
a + O(7^50)
> Valuation(UniformizingElement(L));
1
> Valuation(InertialElement(L));
0
> Eltseq(UniformizingElement(L));
[ 3 + O(7^50), 6 + O(7^50), 1 + O(7^50) ]

```