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**ThePerfectHacker** Here is a field of order 4, $\displaystyle F=\mathbb{Z}_2[t]/(t^2+t+1)$. Therefore, $\displaystyle F = \{ 0,1,\alpha,\alpha+1\}$ where $\displaystyle \alpha = t + (t^2+t+1)$ and $\displaystyle \alpha^2 = \alpha + 1$.

For $\displaystyle x^2 + Nx+1$ to be irreducible it means it cannot have any zeros in $\displaystyle F$. The possible polynomials are: $\displaystyle x^2+1,x^2+x+1,x^2+\alpha x+1, x^2+(\alpha+1)x+1$. Which of these polynomials has zeros in $\displaystyle F$? The ones that do not are the irreducible ones.