6.11.1.4 PolynomialRing_generic Objects

class PolynomialRing_generic
Univariate polynomial ring over a commutative ring.
PolynomialRing_generic( base_ring, [name=False], [sparse=None])

sage: Q = RationalField()
sage: R = PolynomialRing(Q)
sage: x = R.gen()
sage: R(-1) + R(1)
0
sage: (x - Q('2/3'))*(x**2 - 8*x + 16)
x^3 - 26/3*x^2 + 64/3*x - 32/3

Instances of class PolynomialRing_generic have the following methods (in addition to inherited methods and special methods):

base_ring,$  $ characteristic,$  $ cyclotomic_polynomial,$  $ gen,$  $ is_field,$  $ is_sparse,$  $ krull_dimension,$  $ ngens,$  $ parameter,$  $ quotient,$  $ random_element

Further documentation:

cyclotomic_polynomial( n)

The nth cyclotomic polynomial.

sage: R = PolynomialRing(RationalField())
sage: R.cyclotomic_polynomial(8)
x^4 + 1
sage: R.cyclotomic_polynomial(12)
x^4 - x^2 + 1
sage: S = PolynomialRing(FiniteField(7))
sage: S.cyclotomic_polynomial(12)
x^4 + 6*x^2 + 1

gen( [n=0])

If this is R[x], return x.

random_element( degree, [bound=0])

Return a random polynomial.

INPUT:
    degree -- an int
    bound -- an int (default: 0, which tries to spread choice across ring,
if implemented)

OUTPUT:
    Polynomial -- A polynomial such that the coefficient of x^i,
    for i up to degree, are coercisions to the base ring of
    random integers between -bound and bound.

Instances of class PolynomialRing_generic also have the following special methods:

__call__,$  $ __cmp__,$  $ __contains__,$  $ __reduce__,$  $ __repr__,$  $ _coerce_,$  $ _latex_,$  $ _PolynomialRing_generic__set_polynomial_class

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