Local Field Elements

Subsections

• Arithmetic
• Predicates on Elements
• Other Operations on Elements

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.

Example RngLocA_elts (H51E7)

> 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) ]