Module: sage.rings.polynomial_quotient_ring_element
Class: PolynomialQuotientRingElement
self, parent, polynomial, [check=True]) |
Create an element of the quotient of a polynomial ring.
INPUT: parent -- a quotient of a polynomial ring polynomial -- a polynomial check -- bool (optional): whether or not to verify that x is a valid element of the polynomial ring and reduced (mod the modulus).
Functions: charpoly,
fcp,
lift,
list,
matrix,
minpoly,
norm,
trace
self) |
The characteristic polynomial of this element, which is by definition the characteristic polynomial of right multiplication by this element.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 -389*x^2 + 2*x - 5, 'a'); a = S.gen() sage: a.charpoly() x^3 - 389*x^2 + 2*x - 5
self) |
Return the factorization of the characteristic polynomial of this element.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 -389*x^2 + 2*x - 5, 'a'); a = S.gen() sage: a.fcp() (x^3 - 389*x^2 + 2*x - 5) sage: S(1).fcp() (x - 1)^3
self) |
Return lift of this polynomial quotient ring element to the unique equivalent polynomial of degree less than the modulus.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: b = a^2 - 3 sage: b a^2 - 3 sage: b.lift() x^2 - 3
self) |
Return list of the elements of self, of length the same as the degree of the quotient polynomial ring.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 + 2*x - 5, 'a'); a = S.gen() sage: a^10 -134*a^2 - 35*a + 300 sage: (a^10).list() [300, -35, -134]
self) |
The matrix of right multiplication by this element on the power basis for the quotient ring.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 + 2*x - 5, 'a'); a = S.gen() sage: a.matrix() [ 0 1 0] [ 0 0 1] [ 5 -2 0]
self) |
The minimal polynomial of this element, which is by definition the minimal polynomial of right multiplication by this element.
self) |
The norm of this element, which is the norm of the matrix of right multiplication by this element.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 -389*x^2 + 2*x - 5, 'a'); a = S.gen() sage: a.norm() 5
self) |
The trace of this element, which is the trace of the matrix of right multiplication by this element.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3 -389*x^2 + 2*x - 5, 'a'); a = S.gen() sage: a.trace() 389
Special Functions: __add__,
__cmp__,
__div__,
__getitem__,
__int__,
__invert__,
__long__,
__mul__,
__neg__,
__pow__,
__reduce__,
__sub__,
_im_gens_,
_repr_
self, right) |
Return the sum of two polynomial ring quotient elements.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: (a^2 - 4) + (a+2) a^2 + a - 2 sage: int(1) + a a + 1
self, other) |
Compare this element with something else, where equality testing coerces the object on the right, if possible (and necessary).
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: S(1) == 1 True sage: a^3 == 2 True
For the purposes of comparison in SAGE the quotient element
is equal to
. This is because when the comparison
is performed, the right element is coerced into the parent of
the left element, and
coerces to
.
sage: a == x True sage: a^3 == x^3 True sage: x^3 x^3 sage: S(x^3) 2
self, right) |
Return the quotient of two polynomial ring quotient elements.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: (a^2 - 4) / (a+2) a - 2
self) |
Coerce this element to an int if possible.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: int(S(10)) 10 sage: int(a) Traceback (most recent call last): ... TypeError: cannot coerce nonconstant polynomial to int
self) |
Coerce this element to a long if possible.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: long(S(10)) 10L sage: long(a) Traceback (most recent call last): ... TypeError: cannot coerce nonconstant polynomial to long
self, right) |
Return the product of two polynomial ring quotient elements.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: (a^2 - 4) * (a+2) 2*a^2 - 4*a - 6
self, n) |
Return a power of a polynomial ring quotient element.
sage: R = PolynomialRing(Integers(9), 'x'); x = R.gen() sage: S = R.quotient(x^4 + 2*x^3 + x + 2, 'a'); a = S.gen() sage: a^100 7*a^3 + 8*a + 7
self, right) |
Return the difference of two polynomial ring quotient elements.
sage: R, x = PolynomialRing(QQ).objgen() sage: S = R.quotient(x^3-2, 'a'); a = S.gen() sage: (a^2 - 4) - (a+2) a^2 - a - 6 sage: int(1) - a -a + 1
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