Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the field or ring to which the coefficients are considered to belong. For example, the polynomial x2 − 2 is irreducible if the coefficients 1 and −2 are considered as integers, but it factors as if the coefficients are considered as real numbers. One says that "the polynomial x2 − 2 is irreducible over the integers but not over the reals".
A polynomial that is not irreducible is sometimes said to be reducible. However, this term must be used with care, as it may refer to other notions of .
Irreducible polynomials appear naturally in the study of polynomial factorization and algebraic field extensions.
It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of "irreducibility" that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors.
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