With its convenient symbolic representation of algebraic numbers, the Wolfram Language's state-of-the-art algebraic number theory capabilities provide a concrete implementation of one of the historically richest areas of pure mathematics—all tightly integrated with the Wolfram Language's powerful unified environment.

Algebraic Numbers and Representation »

AlgebraicNumber — algebraic number represented in a particular field

Root — represent a root of a polynomial

RootApproximant — root approximation

IsolatingInterval  ▪  MinimalPolynomial  ▪  AlgebraicNumberPolynomial  ▪  ...

AlgebraicIntegerQ  ▪  AlgebraicUnitQ  ▪  RootOfUnityQ

AlgebraicNumberNorm  ▪  AlgebraicNumberTrace  ▪  AlgebraicNumberDenominator

Algebraic Number Fields

ToNumberField — find a common field, or express numbers in a given field

NumberFieldIntegralBasis  ▪  NumberFieldClassNumber  ▪  NumberFieldDiscriminant

NumberFieldRegulator  ▪  NumberFieldSignature

NumberFieldNormRepresentatives  ▪  NumberFieldFundamentalUnits  ▪  NumberFieldRootsOfUnity

Factorization

FactorInteger — factorization of integers

Factor — factorization of polynomials

GaussianIntegers — allow factorization over Gaussian integers

Extension — field extension for number theoretic and polynomial operations

RootReduce — reduce an algebraic number to minimal Root form

ToRadicals — convert to explicit radicals