# Quotient fields¶

class sage.categories.quotient_fields.QuotientFields(s=None)

The category of quotient fields over an integral domain

EXAMPLES:

sage: QuotientFields()
Category of quotient fields
sage: QuotientFields().super_categories()
[Category of fields]

class ElementMethods
denominator()

Constructor for abstract methods

EXAMPLES:

sage: def f(x):
....:     "doc of f"
....:     return 1
sage: x = abstract_method(f); x
<abstract method f at ...>
sage: x.__doc__
'doc of f'
sage: x.__name__
'f'
sage: x.__module__
'__main__'

derivative(*args)

The derivative of this rational function, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

_derivative()

EXAMPLES:

sage: F.<x> = Frac(QQ['x'])
sage: (1/x).derivative()
-1/x^2

sage: (x+1/x).derivative(x, 2)
2/x^3

sage: F.<x,y> = Frac(QQ['x,y'])
sage: (1/(x+y)).derivative(x,y)
2/(x^3 + 3*x^2*y + 3*x*y^2 + y^3)

factor(*args, **kwds)

Return the factorization of self over the base ring.

INPUT:

• *args - Arbitrary arguments suitable over the base ring
• **kwds - Arbitrary keyword arguments suitable over the base ring

OUTPUT:

• Factorization of self over the base ring

EXAMPLES:

sage: K.<x> = QQ[]
sage: f = (x^3+x)/(x-3)
sage: f.factor()
(x - 3)^-1 * x * (x^2 + 1)


Here is an example to show that trac ticket #7868 has been resolved:

sage: R.<x,y> = GF(2)[]
sage: f = x*y/(x+y)
sage: f.factor()
(x + y)^-1 * y * x

gcd(other)

Greatest common divisor

Note

In a field, the greatest common divisor is not very informative, as it is only determined up to a unit. But in the fraction field of an integral domain that provides both gcd and lcm, it is possible to be a bit more specific and define the gcd uniquely up to a unit of the base ring (rather than in the fraction field).

AUTHOR:

EXAMPLES:

sage: R.<x> = QQ['x']
sage: p = (1+x)^3*(1+2*x^2)/(1-x^5)
sage: q = (1+x)^2*(1+3*x^2)/(1-x^4)
sage: factor(p)
(-2) * (x - 1)^-1 * (x + 1)^3 * (x^2 + 1/2) * (x^4 + x^3 + x^2 + x + 1)^-1
sage: factor(q)
(-3) * (x - 1)^-1 * (x + 1) * (x^2 + 1)^-1 * (x^2 + 1/3)
sage: gcd(p,q)
(x + 1)/(x^7 + x^5 - x^2 - 1)
sage: factor(gcd(p,q))
(x - 1)^-1 * (x + 1) * (x^2 + 1)^-1 * (x^4 + x^3 + x^2 + x + 1)^-1
sage: factor(gcd(p,1+x))
(x - 1)^-1 * (x + 1) * (x^4 + x^3 + x^2 + x + 1)^-1
sage: factor(gcd(1+x,q))
(x - 1)^-1 * (x + 1) * (x^2 + 1)^-1

lcm(other)

Least common multiple

In a field, the least common multiple is not very informative, as it is only determined up to a unit. But in the fraction field of an integral domain that provides both gcd and lcm, it is reasonable to be a bit more specific and to define the least common multiple so that it restricts to the usual least common multiple in the base ring and is unique up to a unit of the base ring (rather than up to a unit of the fraction field).

The least common multiple is easily described in terms of the prime decomposition. A rational number can be written as a product of primes with integer (positive or negative) powers in a unique way. The least common multiple of two rational numbers $$x$$ and $$y$$ can then be defined by specifying that the exponent of every prime $$p$$ in $$lcm(x,y)$$ is the supremum of the exponents of $$p$$ in $$x$$, and the exponent of $$p$$ in $$y$$ (where the primes that does not appear in the decomposition of $$x$$ or $$y$$ are considered to have exponent zero).

AUTHOR:

EXAMPLES:

sage: lcm(2/3, 1/5)
2


Indeed $$2/3 = 2^1 3^{-1} 5^0$$ and $$1/5 = 2^0 3^0 5^{-1}$$, so $$lcm(2/3,1/5)= 2^1 3^0 5^0 = 2$$.

sage: lcm(1/3, 1/5) 1 sage: lcm(1/3, 1/6) 1/3

Some more involved examples:

sage: R.<x> = QQ[]
sage: p = (1+x)^3*(1+2*x^2)/(1-x^5)
sage: q = (1+x)^2*(1+3*x^2)/(1-x^4)
sage: factor(p)
(-2) * (x - 1)^-1 * (x + 1)^3 * (x^2 + 1/2) * (x^4 + x^3 + x^2 + x + 1)^-1
sage: factor(q)
(-3) * (x - 1)^-1 * (x + 1) * (x^2 + 1)^-1 * (x^2 + 1/3)
sage: factor(lcm(p,q))
(x - 1)^-1 * (x + 1)^3 * (x^2 + 1/3) * (x^2 + 1/2)
sage: factor(lcm(p,1+x))
(x + 1)^3 * (x^2 + 1/2)
sage: factor(lcm(1+x,q))
(x + 1) * (x^2 + 1/3)

numerator()

Constructor for abstract methods

EXAMPLES:

sage: def f(x):
....:     "doc of f"
....:     return 1
sage: x = abstract_method(f); x
<abstract method f at ...>
sage: x.__doc__
'doc of f'
sage: x.__name__
'f'
sage: x.__module__
'__main__'

partial_fraction_decomposition(decompose_powers=True)

Decomposes fraction field element into a whole part and a list of fraction field elements over prime power denominators.

The sum will be equal to the original fraction.

INPUT:

• decompose_powers – whether to decompose prime power
denominators as opposed to having a single term for each irreducible factor of the denominator (default: True)

OUTPUT:

• Partial fraction decomposition of self over the base ring.

AUTHORS:

EXAMPLES:

sage: S.<t> = QQ[]
sage: q = 1/(t+1) + 2/(t+2) + 3/(t-3); q
(6*t^2 + 4*t - 6)/(t^3 - 7*t - 6)
sage: whole, parts = q.partial_fraction_decomposition(); parts
[3/(t - 3), 1/(t + 1), 2/(t + 2)]
sage: sum(parts) == q
True
sage: q = 1/(t^3+1) + 2/(t^2+2) + 3/(t-3)^5
sage: whole, parts = q.partial_fraction_decomposition(); parts
[1/3/(t + 1), 3/(t^5 - 15*t^4 + 90*t^3 - 270*t^2 + 405*t - 243), (-1/3*t + 2/3)/(t^2 - t + 1), 2/(t^2 + 2)]
sage: sum(parts) == q
True
sage: q = 2*t / (t + 3)^2
sage: q.partial_fraction_decomposition()
(0, [2/(t + 3), -6/(t^2 + 6*t + 9)])
sage: for p in q.partial_fraction_decomposition(): print(p.factor())
(2) * (t + 3)^-1
(-6) * (t + 3)^-2
sage: q.partial_fraction_decomposition(decompose_powers=False)
(0, [2*t/(t^2 + 6*t + 9)])


We can decompose over a given algebraic extension:

sage: R.<x> = QQ[sqrt(2)][]
sage: r =  1/(x^4+1)
sage: r.partial_fraction_decomposition()
(0,
[(-1/4*sqrt2*x + 1/2)/(x^2 - sqrt2*x + 1),
(1/4*sqrt2*x + 1/2)/(x^2 + sqrt2*x + 1)])

sage: R.<x> = QQ[I][]  # of QQ[sqrt(-1)]
sage: r =  1/(x^4+1)
sage: r.partial_fraction_decomposition()
(0, [(-1/2*I)/(x^2 - I), 1/2*I/(x^2 + I)])


We can also ask Sage to find the least extension where the denominator factors in linear terms:

sage: R.<x> = QQ[]
sage: r = 1/(x^4+2)
sage: N = r.denominator().splitting_field('a')
sage: N
Number Field in a with defining polynomial x^8 - 8*x^6 + 28*x^4 + 16*x^2 + 36
sage: R1.<x1>=N[]
sage: r1 = 1/(x1^4+2)
sage: r1.partial_fraction_decomposition()
(0,
[(-1/224*a^6 + 13/448*a^4 - 5/56*a^2 - 25/224)/(x1 - 1/28*a^6 + 13/56*a^4 - 5/7*a^2 - 25/28),
(1/224*a^6 - 13/448*a^4 + 5/56*a^2 + 25/224)/(x1 + 1/28*a^6 - 13/56*a^4 + 5/7*a^2 + 25/28),
(-5/1344*a^7 + 43/1344*a^5 - 85/672*a^3 - 31/672*a)/(x1 - 5/168*a^7 + 43/168*a^5 - 85/84*a^3 - 31/84*a),
(5/1344*a^7 - 43/1344*a^5 + 85/672*a^3 + 31/672*a)/(x1 + 5/168*a^7 - 43/168*a^5 + 85/84*a^3 + 31/84*a)])


Or we may work directly over an algebraically closed field:

sage: R.<x> = QQbar[]
sage: r =  1/(x^4+1)
sage: r.partial_fraction_decomposition()
(0,
[(-0.1767766952966369? - 0.1767766952966369?*I)/(x - 0.7071067811865475? - 0.7071067811865475?*I),
(-0.1767766952966369? + 0.1767766952966369?*I)/(x - 0.7071067811865475? + 0.7071067811865475?*I),
(0.1767766952966369? - 0.1767766952966369?*I)/(x + 0.7071067811865475? - 0.7071067811865475?*I),
(0.1767766952966369? + 0.1767766952966369?*I)/(x + 0.7071067811865475? + 0.7071067811865475?*I)])


We do the best we can over inexact fields:

sage: R.<x> = RealField(20)[]
sage: q = 1/(x^2 + x + 2)^2 + 1/(x-1); q
(x^4 + 2.0000*x^3 + 5.0000*x^2 + 5.0000*x + 3.0000)/(x^5 + x^4 + 3.0000*x^3 - x^2 - 4.0000)
sage: whole, parts = q.partial_fraction_decomposition(); parts
[1.0000/(x - 1.0000), 1.0000/(x^4 + 2.0000*x^3 + 5.0000*x^2 + 4.0000*x + 4.0000)]
sage: sum(parts)
(x^4 + 2.0000*x^3 + 5.0000*x^2 + 5.0000*x + 3.0000)/(x^5 + x^4 + 3.0000*x^3 - x^2 - 4.0000)

xgcd(other)

Return a triple (g,s,t) of elements of that field such that g is the greatest common divisor of self and other and g = s*self + t*other.

Note

In a field, the greatest common divisor is not very informative, as it is only determined up to a unit. But in the fraction field of an integral domain that provides both xgcd and lcm, it is possible to be a bit more specific and define the gcd uniquely up to a unit of the base ring (rather than in the fraction field).

EXAMPLES:

sage: QQ(3).xgcd(QQ(2))
(1, 1, -1)
sage: QQ(3).xgcd(QQ(1/2))
(1/2, 0, 1)
sage: QQ(1/3).xgcd(QQ(2))
(1/3, 1, 0)
sage: QQ(3/2).xgcd(QQ(5/2))
(1/2, 2, -1)

sage: R.<x> = QQ['x']
sage: p = (1+x)^3*(1+2*x^2)/(1-x^5)
sage: q = (1+x)^2*(1+3*x^2)/(1-x^4)
sage: factor(p)
(-2) * (x - 1)^-1 * (x + 1)^3 * (x^2 + 1/2) * (x^4 + x^3 + x^2 + x + 1)^-1
sage: factor(q)
(-3) * (x - 1)^-1 * (x + 1) * (x^2 + 1)^-1 * (x^2 + 1/3)
sage: g,s,t = xgcd(p,q)
sage: g
(x + 1)/(x^7 + x^5 - x^2 - 1)
sage: g == s*p + t*q
True


An example without a well defined gcd or xgcd on its base ring:

sage: K = QuadraticField(5)
sage: O = K.maximal_order()
sage: R = PolynomialRing(O, 'x')
sage: F = R.fraction_field()
sage: x = F.gen(0)
sage: x.gcd(x+1)
1
sage: x.xgcd(x+1)
(1, 1/x, 0)
sage: zero = F.zero()
sage: zero.gcd(x)
1
sage: zero.xgcd(x)
(1, 0, 1/x)
sage: zero.xgcd(zero)
(0, 0, 0)

class ParentMethods
super_categories()

EXAMPLES:

sage: QuotientFields().super_categories()
[Category of fields]