Built-in Mathematica Symbol

PolynomialMod

PolynomialMod[poly, m]
gives the polynomial poly reduced modulo m.

PolynomialMod[poly, {m₁, m₂, ...}]
reduces modulo all of the m_i.

PolynomialMod[poly, m] for integer m gives a polynomial in which all coefficients are reduced modulo m.

When m is a polynomial, PolynomialMod[poly, m] reduces poly by subtracting polynomial multiples of m, to give a result with minimal degree and leading coefficient.

PolynomialMod gives results according to a definite convention; other conventions could yield results differing by multiples of m.

Unlike PolynomialRemainder, PolynomialMod never performs divisions in generating its results.

Reduce a polynomial modulo 2:

Reduce a polynomial modulo another polynomial:

Reduce a polynomial modulo 2:

Reduce a polynomial modulo another polynomial:

Reduce a polynomial modulo an integer:

Reduce a polynomial modulo a polynomial:

Reduce a polynomial modulo a polynomial and an integer:

Reduce a polynomial modulo two polynomials and an integer:

With the default CoefficientDomain->Rationals, integer coefficients can be inverted:

With CoefficientDomain->Integers, PolynomialMod does not invert integer coefficients:

Reduce a polynomial modulo a polynomial over the integers modulo 3:

Reduce all coefficients of a polynomial modulo an integer:

For univariate rational polynomials PolynomialRemainder is the same as PolynomialMod:

PolynomialRemainder considers all polynomials to be univariate in the specified variable:

For multivariate polynomials PolynomialMod picks its own variable order:

The main variable here is a:

PolynomialRemainder considers parameters to be invertible:

PolynomialMod does not invert symbolic expressions: