Question

Find all irreducible polynomials of

(a) degree 2 in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$

Short answer

The only irreducible polynomial is $x^2+x+1$.

Step-by-step solution

Irreducible polynomial

A polynomial is said to be irreducible when its only divisors are its associates and nonzero constant polynomial.

The polynomial of degree 2 is in the form $x^2+a_{1}x+a_0$in ${\mathrm{\mathbb{Z}}}_2$, and the elements in ${\mathrm{\mathbb{Z}}}_2$are $\left\{0,1\right\}$.

Find Irreducible polynomial

Assume that $a_0=0$; then the polynomial is $x^2+a_{1}x=x\left(x+a_1\right)$, which is reducible. This implies that the only choice for $a_0$ is 1, that is, $a_0=1$.

Now, consider $a_0=1$ and $a_1=0$; then the polynomial is $x^2+1=\left(x+1\right)\left(x+1\right)$, which is reducible. So, the only polynomial left is $x^2+x+1$.

Rewrite the polynomial as:

$x^2+x+1=\left(x+1\right)x+1$
Since the obtained polynomial is not divisible by X or $x+1$, it is irreducible.

Hence, the only irreducible polynomial is $x^2+x+1$.