factorability

Factorability is the property of an algebraic expression to be written as a product of nontrivial factors within a specified algebraic structure, such as a ring or field. In the context of polynomials, a polynomial is factorable if it can be expressed as a product of polynomials of smaller positive degree with coefficients in the same coefficient domain; it is irreducible if no such nontrivial factorization exists. Factorability depends on the chosen domain, since a polynomial may factor over one ring but be irreducible over another. For example, x^2 − 5x + 6 factors over the integers as (x − 2)(x − 3), and hence over the rationals as well; x^2 + 1 is irreducible over the real numbers and over the rationals.

Gauss's lemma shows that, for polynomials with integer coefficients, factorization in Z[x] corresponds to factorization in Q[x] up to units. Over the reals, factoring a polynomial means expressing it as a product of real polynomials of smaller degree; over the complex numbers, every polynomial splits into linear factors.

Common methods for testing and finding factors include the rational root theorem, factoring by grouping, and gcd-based approaches. Irreducibility criteria such as Eisenstein's criterion provide sufficient conditions for irreducibility over the rationals. In computational settings, polynomial factorization algorithms exist for various domains, including Berlekamp and Cantor–Zassenhaus for polynomials over finite fields, and LLL-based methods for integer polynomials.

Factorability also contrasts with integer factorization, the decomposition of integers into primes, and with broader notions of primality. In summary, factorability describes whether and how expressions can be decomposed into simpler, nontrivial factors within a given algebraic framework.