Question

Find a fourth-degree polynomial in ${\mathbf{\mathbb{Z}}}_{\mathbf{2}}\left[x\right]$whose roots are the four elements of the field. ${\mathbf{\mathbb{Z}}}_{\mathbf{2}}\mathbf{/}\left(x^2+x+1\right)$, whose tables are given in Example 3. [Hint: The Factor Theorem may be helpful.]

Short answer

$f\left(y\right)=y^4+y\in {\mathrm{\mathbb{Z}}}_2\left[y\right]$is a fourth-degree polynomial.

Step-by-step solution

Determine the fourth polynomial

Let’s consider the polynomial

$f\left(y\right)=y^4+y\in {\mathrm{\mathbb{Z}}}_2\left[y\right]$, in ${\mathrm{\mathbb{Z}}}_2\left[x\right]/\left(x^2+x+1\right)$; then

$$\begin{gathered} f\left(\left[0\right]\right)=\left[0^4+0\right]=\left[0\right]; \\ f\left(\left[1\right]\right)=\left[1^4+1\right]=\left[0\right]; \\ f\left(\left[x\right]\right)=\left[x^4+x\right]=\left[x\left(x+1\right)\left(x^2+x+1\right)\right]=\left[0\right]; \\ f\left(\left[x+1\right]\right)=\left[{\left(x+1\right)}^4+x+1\right]=\left[x\left(x+1\right)\left(x^2+x+1\right)\right]=\left[0\right] \end{gathered}$$

Thus, this is the fourth-degree polynomial.