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an object cannot be expressed as a product of at least two
non-trivial factors in a given set. See also Factorization .

For any Field ''F'', the Ring
of Polynomial s with coefficients in
''F'' is denoted by F {Link without Title} . A polynomial
p(x) in F {Link without Title} 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 {Link without Title} .

This definition depends on the field ''F''. Some simple
examples will be 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 Field s.

It is helpful to compare irreducible polynomials to
Prime Number s: prime numbers (together with the
corresponding negative numbers of equal modulus) are the
irreducible Integer s. They exhibit many of the general properties
of the concept 'irreducibility' that equally apply to irreducible
polynomials, such as the essentially unique factorization into
prime or irreducible factors:

Every polynomial p(x) in F {Link without Title}
can be factorized into polynomials that are irreducible over ''F''.
This factorization is unique Up To Permutation of the
factors and the multiplication of constants from ''F'' to the
factors.


SIMPLE EXAMPLES


The following five polynomials demonstrate some
elementary properties of reducible and irreducible
polynomials:

:p_1(x)=x^2+4x+4\,=(x+2)(x+2),
:p_2(x)=x^2-4\,=(x-2)(x+2),
:p_3(x)=x^2-4/9\,=(x-2/3)(x+2/3),
:p_4(x)=x^2-2\,=(x-\sqrt{2})(x+\sqrt{2}),
:p_5(x)=x^2+1\,=(x-i)(x+i).

Over the ring Z of Integer s,
the first two polynomials are reducible,
but the other three are irreducible.

Over the field Q of Rational Number s,
the first three polynomials are reducible,
but the other two polynomials are irreducible.

Over the field R of Real Number s,
the first four polynomials are reducible, but p_5(x) is still irreducible.

Over the field C of Complex Number s, all
five polynomials are reducible.

In fact over C, every non-constant polynomial can be
factored into linear factors

:p(z)=a_n (z-z_1)(z-z_2)\cdots(z-z_n)

where a_n is the leading coefficient of the polynomial
and z_1,\ldots,z_n are the zeros of p(z).
Hence, all irreducible polynomials are of degree 1.
This is the Fundamental Theorem Of Algebra .

Note: The existence of an essentially ''unique'' factorization
p_5(x)=x^2+1=(x-i)(x+i)
of p_5(x) into factors that do ''not'' belong to
Q {Link without Title} implies that this polynomial is irreducible
over Q:
there cannot be another factorization.

These examples demonstrate the relationship between the zeros of a polynomial (solutions of an algebraic equation)
and the factorization of the polynomial into linear factors.

The existence of irreducible polynomials of degree greater
than one (without zeros in the original field) historically motivated
the Extension of that original number field
so that even these polynomials can be reduced into linear factors:
from rational numbers to real numbers and further to complex
numbers.

For algebraic purposes, the extension from rational numbers to
real numbers is often too 'radical':
It introduces Transcendental Number s (that are not the solutions of
algebraic equations with rational coefficients). These numbers are not
needed for the algebraic purpose of factorizing polynomials (but they
are necessary for the use of real numbers in Analysis ). Thus,
there is a purely algebraic process to Extend
a given field ''F'' with a given polynomial p(x) to a
larger field where this polynomial p(x) can be reduced
into linear factors. The study of such extensions is the starting point
of Galois Theory .


Generalization


If ''R'' is an Integral Domain , an element ''f'' of ''R'' which is neither zero nor a unit is called irreducible if there are no non-units ''g'' and ''h'' with ''f'' = ''gh''. One can show that every Prime element is irreducible; the converse is not true in general but holds in Unique Factorization Domain s. The polynomial ring ''F'' {Link without Title} over a field ''F'' is a unique factorization domain.


SEE ALSO