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Irreducible polynomial

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A polynomial in n variables over a field k that is an irreducible element of the polynomial ring k[x_1,\ldots,x_n], that is, it cannot be represented in the form f=gh where g and h are non-constant polynomials with coefficients in k (irreducibility over k). A polynomial is called absolutely irreducible if it is irreducible over the algebraic closure of its field of coefficients. The absolutely irreducible polynomials of a single variable are the polynomials of degree 1. In the case of several variables there are absolutely irreducible polynomials of arbitrarily high degree, for example, any polynomial of the form f(x_1,\ldots,x_{n-1})+x_n is absolutely irreducible.

The polynomial ring k[x_1,\ldots,x_n] is factorial (cf. Factorial ring): Any polynomial splits into a product of irreducibles and this factorization is unique up to constant factors. Over the field of real numbers any irreducible polynomial in a single variable is of degree 1 or 2 and a polynomial of degree 2 is irreducible if and only if its discriminant is negative. Over an arbitrary algebraic number field there are irreducible polynomials of arbitrarily high degree; for example, x^n+px+p, where n>1 and p is a prime number, is irreducible in \mathbf Q[x] by Eisenstein's criterion (see Algebraic equation).

Let A be an integrally closed ring with field of fractions k and let f(x)\in A[x] be a polynomial in a single variable with leading coefficient 1. If f(x)=g(x)h(x) in k[x] and both g(x) and h(x) have leading coefficient 1, then g(x),h(x)\in A[x] (Gauss' lemma).

Reduction criterion for irreducibility. Let \sigma\colon A\to B be a homomorphism of integral domains. If f(x) and \sigma(f(x)) have the same degree and if \sigma(f(x)) is irreducible over the field of fractions of B, then there is no factorization f(x)=g(x)h(x) where g(x),h(x)\in A[x] and g(x) and h(x) are not constant. For example, a polynomial f(x)\in\mathbf Z[x] with leading coefficient 1 is prime in \mathbf Z[x] (hence irreducible in \mathbf Q[x]) if for some prime p the polynomial \sigma(f(x)) obtained from f(x) by reducing the coefficients modulo p is irreducible.

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] S. Lang, "Algebra" , Addison-Wesley (1974)
[3] O. Zariski, P. Samuel, "Commutative algebra" , 1–2 , Springer (1975)


Comments

A factorial ring is also known as a unique factorization domain (UFD).

How to Cite This Entry:
Irreducible polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_polynomial&oldid=43592
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article