Structure Operations

Contents

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, ..., xn) 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, ..., yn > : FldFunRat, Rng -> Map
hom< P -> S | y1, ..., yn > : FldFunRat, Rng -> Map
Given a function field F=R(x1, ..., xn), a ring S, a map f : F -> S and n elements y1, ..., yn∈S, create the homomorphism g : F -> S by applying the rules of g(rx1a1 ... xnan)=f(r)y1a1 ... ynan 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 1R to 1S. Also, the images yi are allowed to be from a structure that allows automatic coercion into S.

Example FldFunRat_Homomorphism (H44E2)

In this example we map Q(x, y) into the number field Q(root 3 of 2, Sqrt(5)) by sending x to root 3 of 2 and y to Sqrt(5) and the identity map on the coefficients (which we omit). 
> 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
V2.28, 13 July 2023