Definition:Irreducible Polynomial
Definition
Definition 1
Let $R$ be an integral domain.
An irreducible polynomial over $R$ is an irreducible element of the polynomial ring $R \left[{X}\right]$.
Definition 2: for fields
Let $K$ be a field.
An irreducible polynomial over $K$ is a nonconstant polynomial over $K$ that is not the product of two polynomials of smaller degree.
Definition 3: for fields
Let $K$ be a field.
An irreducible polynomial over $K$ is a polynomial over $K$ that is not the product of two nonconstant polynomials.
Examples
$x^2 - 2$ in Ring of Polynomials over Reals
Consider the polynomial:
- $\map P x = x^2 - 2$
over the ring of polynomials $\R \sqbrk X$ over the real numbers.
Then $\map P x$ is not irreducible, as from Difference of Two Squares:
- $x^2 - 2 \equiv \paren {x + \sqrt 2} \paren {x - \sqrt 2} $
$x^2 - 2$ in Ring of Polynomials over Rationals
Consider the polynomial:
- $\map P x = x^2 - 2$
over the ring of polynomials $\Q \sqbrk X$ over the rational numbers.
Then $\map P x$ is irreducible.
$X^2 + 1$ in Ring of Polynomials over Reals
Let $\R \sqbrk X$ be the ring of polynomials in $X$ over the real numbers $\R$.
Then the polynomial $X^2 + 1$ is an irreducible element of $\R \sqbrk X$.
$x^2 + 1$ in Ring of Polynomials over Complex Numbers
Consider the polynomial:
- $\map P x = x^2 + 1$
over the ring of polynomials $\C \sqbrk X$ over the complex numbers.
Then $\map P x$ is not irreducible, as:
- $x^2 + 1 \equiv \paren {x + i} \paren {x - i}$
Also see
A part of this page has to be extracted as a theorem. |
- By Units of Ring of Polynomial Forms over Field, a polynomial in a single indeterminate with coefficients in a field is irreducible if and only if it is not a product of two polynomial forms of smaller degree.
This is not necessarily true for polynomials over a commutative ring.
- Results about irreducible polynomials can be found here.