Module: sage.rings.quotient_ring_element
Quotient Ring Elements
Author: William Stein
TODO: This implementation is very basic.
Class: QuotientRingElement
self, parent, rep, [reduce=True]) |
An element of a quotient ring
.
sage: R.<x> = PolynomialRing(ZZ) sage: S.<xbar> = R.quo((4 + 3*x + x^2, 1 + x^2)); S Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal (x^2 + 3*x + 4, x^2 + 1) sage: v = S.gens(); v (xbar,)
sage: loads(v[0].dumps()) == v[0] True
sage: R.<x,y> = PolynomialRing(QQ, 2) sage: S = R.quo(x^2 + y^2); S Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2) sage: S.gens() (xbar, ybar)
We name each of the generators.
sage: S.<a,b> = R.quotient(x^2 + y^2) sage: a a sage: b b sage: a^2 + b^2 == 0 True sage: b.lift() y sage: (a^3 + b^2).lift() -x*y^2 + y^2
Functions: copy,
is_unit,
lc,
lift,
lm,
lt,
monomials,
reduce,
variables
self) |
Return the leading coefficent of this quotient ring element.
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex') sage: I = sage.rings.ideal.FieldIdeal(R) sage: Q = R.quo( I ) sage: f = Q( z*y + 2*x ) sage: f.lc() 2
self) |
Return the leading monomial of this quotient ring element.
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex') sage: I = sage.rings.ideal.FieldIdeal(R) sage: Q = R.quo( I ) sage: f = Q( z*y + 2*x ) sage: f.lm() xbar
self) |
Return the leading term of this quotient ring element.
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex') sage: I = sage.rings.ideal.FieldIdeal(R) sage: Q = R.quo( I ) sage: f = Q( z*y + 2*x ) sage: f.lt() 2*xbar
self, G) |
Reduce this quotient ring element by a set of quotient ring elements G.
Input:
sage: P.<a,b,c,d,e> = PolynomialRing(GF(2), 5, order='lex') sage: I1 = ideal([a*b + c*d + 1, a*c*e + d*e, a*b*e + c*e, b*c + c*d*e + 1]) sage: Q = P.quotient( sage.rings.ideal.FieldIdeal(P) ) sage: I2 = ideal([Q(f) for f in I1.gens()]) sage: f = Q((a*b + c*d + 1)^2 + e) sage: f.reduce(I2.gens()) ebar
Special Functions: __cmp__,
__float__,
__init__,
__int__,
__invert__,
__long__,
__neg__,
__pos__,
__rdiv__,
_add_,
_div_,
_integer_,
_magma_,
_mul_,
_rational_,
_reduce_,
_repr_,
_singular_,
_sub_
self, [magma=None]) |
Returns the MAGMA representation of this quotient ring element.
sage: P.<x,y> = PolynomialRing(GF(2)) sage: Q = P.quotient(sage.rings.ideal.FieldIdeal(P)) sage: xbar, ybar = Q.gens() sage: xbar._magma_() # optional requires magma x
self, [singular=Singular]) |
Return Singular representation of self.
Input:
sage: P.<x,y> = PolynomialRing(GF(2),2) sage: I = sage.rings.ideal.FieldIdeal(P) sage: Q = P.quo(I) sage: Q._singular_() // characteristic : 2 // number of vars : 2 // block 1 : ordering dp // : names x y // block 2 : ordering C // quotient ring from ideal _[1]=x2+x _[2]=y2+y sage: xbar = Q(x); xbar xbar sage: xbar._singular_() x sage: Q(xbar._singular_()) # a round-trip xbar
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