Module polynomial_quotient_ring_element

Elements of Quotients of Univariate Polynomial Rings

EXAMPLES:
We create a quotient of a univariate polynomial ring over $\ZZ$.
    sage: R.<x> = ZZ[]
    sage: S.<a> = R.quotient(x^3 + 3*x -1)
    sage: 2 * a^3
    -6*a + 2

Next we make a univeriate polynomial ring over $\Z[x]/(x^3+3x-1)$.
    sage: S.<y> = S[]

And, we quotient out that by $y^2 + a$.
    sage: T.<z> = S.quotient(y^2+a)

In the quotient $z^2$ is $-a$.
    sage: z^2
    -a

And since $a^3 = -3x + 1$, we have:
    sage: z^6
    3*a - 1

    sage: R.<x> = PolynomialRing(Integers(9))
    sage: S.<a> = R.quotient(x^4 + 2*x^3 + x + 2)
    sage: a^100
    7*a^3 + 8*a + 7

    sage: R.<x> = PolynomialRing(QQ)
    sage: S.<a> = R.quotient(x^3-2)             
    sage: a
    a
    sage: a^3
    2

For the purposes of comparison in SAGE the quotient element
$a^3$ is equal to $x^3$.  This is because when the comparison
is performed, the right element is coerced into the parent of
the left element, and $x^3$ coerces to $a^3$.

    sage: a == x
    True
    sage: a^3 == x^3
    True
    sage: x^3
    x^3
    sage: S(x^3)
    2


AUTHOR:
    -- William Stein