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Common Divisors and Common Multiples

Common Divisors and Common Multiples

The functions in this section are restricted to univariate polynomials over a field, over the integers, or over a residue class ring of integers with prime modulus, or any polynomial ring over these.

Subsections

Common Divisors and Common Multiples

GreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
Gcd(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
GCD(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
The greatest common divisor of polynomials f and g in a univariate polynomial ring P. The coefficient ring of P must be either a field, the ring of integers or a residue class ring with prime modulus, or a polynomial ring over any of these.
ExtendedGreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt, RngUPolElt
Xgcd(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt, RngUPolElt
XGCD(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt, RngUPolElt
The extended greatest common divisor of polynomials f and g in a univariate polynomial ring P: the function returns polynomials d, u and v in P such that d is the greatest common divisor of f and g, and u.f + v.g = d. The coefficient ring of P must be either a field, or a residue class ring with prime modulus.
LeastCommonMultiple(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
Lcm(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
LCM(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
The least common multiple of polynomials f and g in a univariate polynomial ring P. The coefficient ring of P must be either a field, the ring of integers or a residue class ring with prime modulus.

Content and Primitive Part

Content(p) : RngUPolElt -> RngIntElt
The content of p, that is, the greatest common divisor of the coefficients of p as an element of the coefficient ring.
PrimitivePart(p) : RngUPolElt -> RngUPolElt
The primitive part of p, being p divided by the content of p.
ContentAndPrimitivePart(p) : RngUPolElt -> RngIntElt, RngUPolElt
Contpp(p) : RngUPolElt -> RngIntElt, RngUPolElt
The content (the greatest common divisor of the coefficients) of p, as an element of the coefficient ring, as well as the primitive part (p divided by the content) of p.
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