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Algebraic Number Theory
With its convenient symbolic representation of algebraic numbers, Mathematica'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 Mathematica's powerful unified environment.
AlgebraicNumber algebraic number represented in a particular field
Root represent a root of a polynomial
RootApproximant root approximation

Algebraic Number Fields
ToNumberField find a common field, or express numbers in a given field
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