Question

Find all irreducible polynomials of

(b) degree 3 in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$

Short answer

The irreducible polynomials of degree 3 in ${\mathrm{\mathbb{Z}}}_2\left[x\right]$ are$x^3+x^2+1$ and $x^3+x+1$.

Step-by-step solution

Irreducible polynomial 

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

The polynomial of degree 3 are in the form $x^3+a_2{x}^2+a_{1}x+a_0$ 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$x^3+a_2{x}^2+a_{1}x=x\left(x^2+a_{2}x+a_1\right)$ , which is reducible. This implies that the only choice for $a_0$ is 1, that is,$a_0=1$ .

Now, consider $a_0=1$, $a_1=0$ and $a_2=0$; then the polynomial is $x^3+1=\left(x^2+x+1\right)\left(x+1\right)$, which is reducible.

Now, consider $a_0=1$, $a_1=1$ and $a_2=1$, then the polynomial is $x^3+x^2+x+1={\left(x+1\right)}^3$, which is reducible. This implies that only two irreducible polynomials of degree 3 are possible, that is, $x^3+x^2+1$ and $x^3+x+1$

Rewrite the polynomials as:

$$\begin{gathered} x^3+x^2+1=\left(x+1\right)x^2+1 \\ {x}^3+x+1=\left(x+1\right)\left(x^2+x\right)+1 \end{gathered}$$
Since the obtained polynomial is not divisible by x or $x+1$, it is irreducible.

Hence, the irreducible polynomials are $x^3+x^2+1$and $x^3+x+1$.