High Quality Content by WIKIPEDIA articles! In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization. For any field F, the ring of polynomials with coefficients in F is denoted by F[x]. A polynomial p(x) in F[x] is called irreducible over F if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from F[x]. The property of irreducibility depends on the field F; a polynomial may be irreducible over some fields but reducible over others. Some simple examples are discussed below. Galois theory studies the relationship between a field, its Galois group, and its irreducible polynomials in depth. Interesting and non-trivial applications can be found in the study of finite fields.

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