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 univariate polynomial ring over \(\ZZ[x]/(x^3+3x-1)\).

sage: S1.<y> = S[]

And, we quotient out that by \(y^2 + a\).

sage: T.<z> = S1.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

AUTHORS:

  • William Stein
class sage.rings.polynomial.polynomial_quotient_ring_element.PolynomialQuotientRingElement(parent, polynomial, check=True)

Bases: sage.rings.polynomial.polynomial_singular_interface.Polynomial_singular_repr, sage.structure.element.CommutativeRingElement

Element of a quotient of a polynomial ring.

EXAMPLES:

sage: P.<x> = QQ[]
sage: Q.<xi> = P.quo([(x^2+1)])
sage: xi^2
-1
sage: singular(xi)
xi
sage: (singular(xi)*singular(xi)).NF('std(0)')
-1
charpoly(var)

The characteristic polynomial of this element, which is by definition the characteristic polynomial of right multiplication by this element.

INPUT:

  • var - string - the variable name

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quo(x^3 -389*x^2 + 2*x - 5)
sage: a.charpoly('X')
X^3 - 389*X^2 + 2*X - 5
fcp(var='x')

Return the factorization of the characteristic polynomial of this element.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5)
sage: a.fcp('x')
x^3 - 389*x^2 + 2*x - 5
sage: S(1).fcp('y')
(y - 1)^3
field_extension(names)

Given a polynomial with base ring a quotient ring, return a 3-tuple: a number field defined by the same polynomial, a homomorphism from its parent to the number field sending the generators to one another, and the inverse isomorphism.

INPUT:

  • names - name of generator of output field

OUTPUT:

  • field
  • homomorphism from self to field
  • homomorphism from field to self

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<alpha> = R.quotient(x^3-2)
sage: F.<a>, f, g = alpha.field_extension()
sage: F
Number Field in a with defining polynomial x^3 - 2
sage: a = F.gen()
sage: f(alpha)
a
sage: g(a)
alpha

Over a finite field, the corresponding field extension is not a number field:

sage: R.<x> = GF(25,'b')['x']
sage: S.<a> = R.quo(x^3 + 2*x + 1)
sage: F.<b>, g, h = a.field_extension()
sage: h(b^2 + 3)
a^2 + 3
sage: g(x^2 + 2)
b^2 + 2

We do an example involving a relative number field:

sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: Q.<b> = S.quo(X^3 + 2*X + 1)
sage: F, g, h = b.field_extension('c')

Another more awkward example:

sage: R.<x> = QQ['x']
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K['X']
sage: f = (X+a)^3 + 2*(X+a) + 1
sage: f
X^3 + 3*a*X^2 + (3*a^2 + 2)*X + 2*a + 3
sage: Q.<z> = S.quo(f)
sage: F.<w>, g, h = z.field_extension()
sage: c = g(z)
sage: f(c)
0
sage: h(g(z))
z
sage: g(h(w))
w

AUTHORS:

  • Craig Citro (2006-08-06)
  • William Stein (2006-08-06)
is_unit()

Return True if self is invertible.

Warning

Only implemented when the base ring is a field.

EXAMPLES:

sage: R.<x> = QQ[]
sage: S.<y> = R.quotient(x^2 + 2*x + 1)
sage: (2*y).is_unit()
True
sage: (y+1).is_unit()
False

TESTS:

Raise an exception if the base ring is not a field (see trac ticket #13303):

sage: Z16x.<x> = Integers(16)[]
sage: S.<y> =  Z16x.quotient(x^2 + x + 1)
sage: (2*y).is_unit()
Traceback (most recent call last):
...
NotImplementedError: The base ring (=Ring of integers modulo 16) is not a field
lift()

Return lift of this polynomial quotient ring element to the unique equivalent polynomial of degree less than the modulus.

EXAMPLES:

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

Return list of the elements of self, of length the same as the degree of the quotient polynomial ring.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3 + 2*x - 5)
sage: a^10
-134*a^2 - 35*a + 300
sage: (a^10).list()
[300, -35, -134]
matrix()

The matrix of right multiplication by this element on the power basis for the quotient ring.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3 + 2*x - 5)
sage: a.matrix()
[ 0  1  0]
[ 0  0  1]
[ 5 -2  0]
minpoly()

The minimal polynomial of this element, which is by definition the minimal polynomial of right multiplication by this element.

norm()

The norm of this element, which is the norm of the matrix of right multiplication by this element.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5)
sage: a.norm()
5
trace()

The trace of this element, which is the trace of the matrix of right multiplication by this element.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5)
sage: a.trace()
389

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