Irreducible Polynomial

irreducible polynomial [‚ir·ə′dü·sə·bəl ‚päl·ə′nō·mē·əl]

(mathematics)

A polynomial is irreducible over a fieldKif it cannot be written as the product of two polynomials of lesser degree whose coefficients come fromK. Also known as irreducible function.

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Irreducible Polynomial 

a polynomial that cannot be factored into factors of lower degree. The possibility of factoring a polynomial into factors and the irreducibility property depend on the numbers that can be coefficients of the polynomial. Thus, the polynomial x³ + 2 is irreducible if only rational numbers are admitted as coefficients; but it is factored into the product of two irreducible polynomials

if any real numbers can be taken as the coefficients and into the product of three factors

if complex numbers are the coefficients. In the general case, the concept of irreducibility is defined for polynomials with coefficients belonging to an arbitrary field. A polynomial with rational coefficients that cannot be factored into factors of lower degree with rational coefficients is often called an irreducible polynomial.

REFERENCE

Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow, 1968.