Irreducible Polynomials §

Definition §

An irreducible polynomial is a non-constant polynomial that cannot be expressed as the product of two non-constant polynomials. It’s the polynomial equivalent of a prime number in the integers.

In mathematical terms, a polynomial $p(x)$ over a field $F$ is said to be irreducible over $F$ if it cannot be written as the product of two non-constant polynomials from $F[x]$ (note). If such a decomposition is possible, we say $p(x)$ is reducible.

Note: constant polynomials are neither reducible nor irreducible (by convention), whilst polynomials of degree 1 are always irreducible, trivially.

It is of course also important to note which field the irreducibility is being tested, for example $x^2+1$ is irreducible in the real numbers, but reducible to $(x+i)(x-i)$ in the complex numbers.

Importance §

Irreducibility is a central concept in polynomial factorization, algebraic number theory, and several areas of algebra. It helps us understand the structure and properties of polynomials and their roots.

Test for Irreducibility §

There are several methods to test for irreducibility:

Quadratic Polynomials: Given an order 2 polynomial

$$p(x)=a{x}^2+bx+c$$

with integer coefficients, the polynomial is reducible precisely when $b^2-4ac$ is a square in $F$.

Rational Root Theorem: Given a polynomial $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_2{x}^2+a_{1}x+a_0$ with integer coefficients, if the polynomial has a rational root $\frac{p}{q}$, then $p$ divides $a_0$ and $q$ divides $a_n$.

Example §

Consider the polynomial $p(x)=x^2-2$.

Using the Rational Root Theorem, the only potential rational roots are ±1 and ±2. Evaluating the polynomial at these values, we see that none of them are roots. Thus, $p(x)=x^2-2$ doesn’t have any rational roots. It also turns out to be irreducible over the rational numbers, but its roots are $\sqrt{2}$ and $-\sqrt{2}$, which are irrational.

This can be checked by using the discriminant : $b^2-4ac⟹0-4\cdot 1\cdot (-2)⟹8>0\therefore \text{no rational roots.}$

Exercise §

Check if the polynomial $q(x)=x^3-3x+2$ is reducible over the rationals.

Conclusion §

Irreducible polynomials play a fundamental role in understanding the structure of polynomial equations. Being able to determine whether a polynomial is irreducible is an important skill in higher-level mathematics. Always remember to test for potential roots using tools like the Rational Root Theorem, but also be aware that this is just one step in determining irreducibility.