For any field F, the ring of polynomials with coefficients in F is denoted by . A polynomial
in is calledirreducible over , if it is non-constant and cannot be represented as the product of two or more non-constant polynomials from .
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 fields.
It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal modulus) are the irreducible integers. 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 in 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.
The following three polynomials demonstrate some
elementary properties of reducible and irreducible
polynomials:
Simple examples
Over the field Q of rational numbers,
the first polynomial is reducible,
but the other two polynomials are irreducible.
Over the field R of real numbers, the two polynomials and are reducible, but is still irreducible.
Over the field C of complex numbers, all three polynomials are reducible.
In fact over C, every polynomial can be factored into linear factors
Note: The existence of an essentially unique factorization
of into factors that do not belong to
implies that this polynomial is irreducibleover 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 numbers (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 to a
larger field where this polynomial can be reduced
into linear factors. The study of such extensions is the starting point
of Galois theory.
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 domains. The polynomial ring F[x] over a field F is a unique factorization domain.Generalization