factorizable

Factorizable is an adjective used in mathematics to describe objects that can be written as a product of two or more factors of the same kind. Whether an object is considered factorizable depends on the algebraic setting and on what counts as a trivial factor, typically a unit or invertible element. If no nontrivial factorization exists, the object is described as irreducible or prime in its context.

In ring theory and number theory, a nonzero nonunit element is factorizable if it can be expressed as a product of nonunit elements. Factorizations into irreducibles are central to understanding the structure of rings, and in a unique factorization domain this decomposition is unique up to order and associates. For example, 60 factors as 2 × 2 × 3 × 5 in the integers. Over other rings, factorizations can behave differently, and irreducibility can be context-dependent.

In polynomial rings, a polynomial is factorizable over a given field if it can be written as a product of polynomials of smaller degree. Polynomials that cannot be factored are irreducible. Over the complex numbers, every polynomial factors completely into linear factors, while over the real numbers some remain irreducible (for example, x^2 + 1).

In linear algebra and operator theory, factorizations express an object as a product of simpler ones. Matrix factorizations include LU, QR, Cholesky, and singular value decompositions; their existence depends on properties such as nonzero leading minors, symmetry, or positive definiteness. Factorizability also appears in the factorization of differential or linear operators.

Overall, factorizability describes the potential to decompose an object into simpler components, with precise meaning tied to the algebraic structure being considered.