Polynomial algorithms are at the core of classical "computer algebra". Incorporating methods that span from antiquity to the latest cutting-edge research at Wolfram Research, the Wolfram Language has the world's broadest and deepest integrated web of polynomial algorithms. Carefully tuned strategies automatically select optimal algorithms, allowing large-scale polynomial algebra to become a routine part of many types of computations.

Polynomial Elements

Coefficient  ▪  CoefficientList  ▪  CoefficientRules  ▪  Exponent  ▪  Variables

Basic Structural Operations

Expand  ▪  Collect  ▪  MonomialList

Polynomial Factoring & Decomposition »

Factor  ▪  FactorList  ▪  Decompose  ▪  SymmetricReduction  ▪  PolynomialSumOfSquaresList  ▪  ...

Polynomial Division »

PolynomialQuotient  ▪  PolynomialGCD  ▪  PolynomialReduce  ▪  ...

Polynomial Systems »

Solve — find generic solutions for variables

Eliminate — eliminate variables between equations

Resolve — eliminate general quantifiers

Reduce — reduce systems of equations and inequalities to canonical form

SolveValues, NSolveValues — directly gives solution vectors

AsymptoticSolve — asymptotic approximation to algebraic equations

Discriminant  ▪  Resultant  ▪  GroebnerBasis  ▪  CylindricalDecomposition  ▪  CylindricalDecompositionFunction  ▪  ...

Finite Fields

FiniteField — represent a finite field

FiniteFieldElement — represent an element of a finite field

Modulus — specify a modulus

PolynomialMod — reduce coefficients in a polynomial

FiniteFieldEmbedding  ▪  IrreduciblePolynomialQ  ▪  PrimitivePolynomialQ  ▪  ...

Algebraic Number Fields »

GaussianIntegers — do operations over Gaussian integers

Extension — specify a general algebraic extension field

Root — general representation of a polynomial root

MinimalPolynomial — minimal polynomial for a general algebraic number

RootSum  ▪  RootReduce  ▪  ToRadicals  ▪  Cyclotomic  ▪  SymmetricPolynomial  ▪  ...