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
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.
Other Operations on Elements
- 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: Uniformizing and Inertial Elements (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) ]