Structure Operations

Related Structures

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 \(h\).

Category(R): FldFunRat -> Cat

Parent(R): FldFunRat -> Pow

PrimeRing(R): FldFunRat -> Rng

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 \(R(x_1, \ldots, x_n)\) 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, ..., yₙ >: FldFunRat, Rng -> Map

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

Given a function field \(F=R(x_1,\ldots, x_n)\), a ring \(S\), a map \(f : F\rightarrow S\) and \(n\) elements \(y_1, \ldots, y_n\in S\), create the homomorphism \(g : F\rightarrow S\) by applying the rules of \(g(rx_1^{a_1}\cdots x_n^{a_n})=f(r)y_1^{a_1}\cdots y_n^{a_n}\) 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 \(S\).

Example: Homomorphism (ex-bca4bf)

In this example we map \({\mathbb{Q}}(x, y)\) into the number field \({\mathbb{Q}}(\root 3 \of 2, \sqrt{5})\) 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

Run in calculatorCopy input