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 :

> 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) ]

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