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.