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The main structure related to a polynomial ring is its
coefficient ring. Multivariate polynomial rings belong to the
Magma category RngMPol.
Category(P) : RngMPol -> Cat
Parent(P) : RngMPol -> Pow
PrimeRing(P) : RngMPol -> Rng
CoefficientRing(P) : RngMPol -> Rng
Return the coefficient ring of polynomial ring P.
Note that the # operator only returns a value for
finite (quotients of) polynomial rings.
Characteristic(P) : RngMPol -> RngIntElt
#P : RngMPol -> RngIntElt
Return the number of indeterminates of polynomial ring P over its
coefficient ring.
The usual ring functions returning Boolean values are available
on polynomial rings.
IsCommutative(P) : RngMPol -> BoolElt
IsUnitary(P) : RngMPol -> BoolElt
IsFinite(P) : RngMPol -> BoolElt
IsOrdered(P) : RngMPol -> BoolElt
IsField(P) : RngMPol -> BoolElt
IsEuclideanDomain(P) : RngMPol -> BoolElt
IsPID(P) : RngMPol -> BoolElt
IsUFD(P) : RngMPol -> BoolElt
IsDivisionRing(P) : RngMPol -> BoolElt
IsEuclideanRing(P) : RngMPol -> BoolElt
IsDomain(P) : RngMPol -> BoolElt
IsPrincipalIdealRing(P) : RngMPol -> BoolElt
P eq Q : RngMPol, RngMPol -> BoolElt
P ne Q : RngMPol, RngMPol -> BoolElt
The ChangeRing function enables the changing of the coefficient ring
of a polynomial ring.
Given a polynomial ring P=R[x1, ..., xn] of
rank n with coefficient ring R, together with a ring S, construct
the polynomial ring Q=S[x1, ..., xn].
It is necessary that all elements of the old coefficient ring R
can be automatically coerced into the new coefficient ring S.
In its general form, a ring homomorphism taking a polynomial ring 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 | y1, ..., yn > : RngMPol, Rng -> Map
Given a polynomial ring P=R[x1, ..., xn], a ring S, a map
f : R -> S and n elements y1, ..., yn∈S, create
the homomorphism g : P -> S by applying the rules
that g(rx1a1 ... xnan)=f(r)y1a1 ... ynan
for monomials and linearity, that is, g(M + N)=g(M) + g(N).
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.
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();
> R<x, y> := PolynomialRing(Q, 2);
> A<a> := PolynomialRing(IntegerRing());
> N<z, w> := NumberField([a^3-2, a^2+5]);
> h := hom< R -> N | z, w >;
> h(x^11*y^3-x+4/5*y-13/4);
-40*w*z^2 - z + 4/5*w - 13/4
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