Structure Operations

Related Structures

Category(R) : FldFunRat -> Cat

Parent(R) : FldFunRat -> Pow

PrimeRing(R) : FldFunRat -> Rng

IntegerRing(F) : FldFunRat -> RngPol

RingOfIntegers(F) : FldFunRat -> RngPol

Given the rational function field F this returns the polynomial ring from which F was constructed as its field of fractions.

BaseRing(F) : FldFunRat -> Rng

CoefficientRing(F) : FldFunRat -> Rng

The coefficient ring of the (ring of integers of) the rational function field F.

Rank(F) : FldFunRat -> RngIntElt

The rank (number of indeterminates) of the rational function field F.

ValuationRing(F) : FldFunRat -> RngVal

Given the rational function field F for which the coefficients come from a field, this returns the valuation ring of F with respect to the valuation given by the degree. This valuation ring consists of those rational functions g/h for which the degree of h is greater than or equal to that of g.

ValuationRing(F, f) : FldFunRat, RngUPolElt -> RngVal

Given the rational function field F for which the coefficients come from a field, and an irreducible polynomial f in the ring of integers of F, this returns the valuation ring of F with respect to the valuation associated with f. This valuation ring consists of those rational functions g/h for which f divides g but not h.

Invariants

Characteristic(F) : FldFunRat -> FldFunRatElt

Ring Predicates and Booleans

IsCommutative(F) : FldFunRat -> BoolElt

IsUnitary(F) : FldFunRat -> BoolElt

IsFinite(F) : FldFunRat -> BoolElt

IsOrdered(F) : FldFunRat -> BoolElt

IsField(F) : FldFunRat -> BoolElt

IsEuclideanDomain(F) : FldFunRat -> BoolElt

IsPID(F) : FldFunRat -> BoolElt

IsUFD(F) : FldFunRat -> BoolElt

IsDivisionRing(F) : FldFunRat -> BoolElt

IsEuclideanRing(F) : FldFunRat -> BoolElt

IsPrincipalIdealRing(F) : FldFunRat -> BoolElt

IsDomain(F) : FldFunRat -> BoolElt

F eq G : FldFunRat, Rng -> BoolElt

F ne G : FldFunRat, Rng -> BoolElt

Homomorphisms

In its general form a ring homomorphism taking a function field R(x₁, ..., x_n) as domain requires n + 1 pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring R together with the images of the n indeterminates.

hom< P -> S | f, y₁, ..., y_n > : FldFunRat, Rng -> Map

hom< P -> S | y₁, ..., y_n > : FldFunRat, Rng -> Map

Given a function field F=R(x₁, ..., x_n), a ring S, a map f : F -> S and n elements y₁, ..., y_n∈S, create the homomorphism g : F -> S by applying the rules of g(rx₁^a₁ ... x_n^a_n)=f(r)y₁^a₁ ... y_n^a_n for monomials, linearity for polynomials, i.e., g(M + N)=g(M) + g(N), and division for fractions, i.e., g(n/d)=g(n)/g(d).

The coefficient ring map may be omitted, in which case the coefficients are mapped into S by the unitary homomorphism sending 1_R to 1_S. Also, the images y_i are allowed to be from a structure that allows automatic coercion into S.

> Q := RationalField();
> F<x, y> := FunctionField(Q, 2);
> A<a> := PolynomialRing(IntegerRing());
> N<z, w> := NumberField([a^3-2, a^2+5]);
> h := hom< F -> N | z, w >;
> h(x^11*y^3-x+4/5*y-13/4);
-40*w*z^2 - z + 4/5*w - 13/4
> h(x/3);                  
1/3*z
> h(1/x);
1/2*z^2
> 1/z;
1/2*z^2