Irreducible polynomial Totally Explained

Everything about Irreducible Polynomial totally explained

In mathematics, the adjective irreducible means that an object can't 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 polynomials with coefficients in F is denoted by

F[x]

. A polynomial

p(x)

F[x]

is called irreducible over $F$ if it's non-constant and can't be represented as the product of two or more non-constant polynomials from

F[x]

.
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

p(x)

F[x]

can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by constants from F.

Simple examples

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

p_1(x)=x^2+4x+4,=

but factors into 4 linear factors or 2 quadratic factors mod any prime p.

Further Information

Get more info on 'Irreducible Polynomial'.

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