factorizations

Factorization is the process of expressing an object as a product of smaller objects, called factors, that multiply to give the original. In number theory, factorization typically refers to decomposing an integer into a product of integers greater than one. The most familiar case is prime factorization, writing a number as a product of primes. The fundamental theorem of arithmetic states that every integer greater than one has a unique factorization into primes up to the order of the factors and a sign.

In algebra, factorization extends to polynomials. A polynomial is factorized by expressing it as a product of polynomials of lower degree with coefficients in a given field or ring. A polynomial is factorizable if it can be written as a product of irreducible polynomials over that domain. Over the rational numbers, many polynomials factor using Gauss’s lemma and algorithms such as Berlekamp’s algorithm or Cantor–Zassenhaus; over finite fields similar methods exist.

Factorization also occurs in broader algebraic settings. An integral domain in which every element factors uniquely into irreducibles is called a unique factorization domain (UFD); a principal ideal domain (PID) is an even stronger condition that implies unique factorization under certain circumstances. In non-UFDs, factorization need not be unique, as illustrated by classic examples.

Factorization has practical applications in cryptography, computer algebra, coding theory, and the solution of Diophantine equations, among others. Understanding when and how objects factor and whether the factorization is unique is a central theme across number theory and algebra.