Irreducible polynomial
A polynomial in
variables over a field
that is a prime element of the ring
, that is, it cannot be represented in the form
where
and
are non-constant polynomials with coefficients in
(irreducibility over
). 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
is absolutely irreducible.
The polynomial ring 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,
, where
and
is a prime number, is irreducible in
by Eisenstein's criterion (see Algebraic equation).
Let be an integrally closed ring with field of fractions
and let
be a polynomial in a single variable with leading coefficient 1. If
in
and both
and
have leading coefficient 1, then
(Gauss' lemma).
Reduction criterion for irreducibility. Let be a homomorphism of integral domains. If
and
have the same degree and if
is irreducible over the field of fractions of
, then there is no factorization
where
and
and
are not constant. For example, a polynomial
with leading coefficient 1 is prime in
(hence irreducible in
) if for some prime
the polynomial
obtained from
by reducing the coefficients modulo
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).
Irreducible polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_polynomial&oldid=31615