Let $\mathbb{F}=\{0,1,\alpha,\alpha+1\}$ be a field with operations $+$ and $\cdot$. Determine all irreducible polynomials over $\mathbb{F}$ of degree at most $3$.
In order to find all irreducible polynomials has degree at least $3$, I think I need to make two tables, one is addition and one is multiplication \begin{array}{c|cccc} + & 0 & 1 & \alpha & \alpha+1\\ \hline 0 & 0 & 1 & \alpha & \alpha+1 \\ 1 & 1 & 0 & \alpha+1 & \alpha \\ \alpha & \alpha & \alpha+1 & 0 & 1 \\ \alpha+1 & \alpha+1 & \alpha & 1 & 0 \\ \end{array}
\begin{array}{c|cccc} \times & 0 & 1 & \alpha & \alpha+1\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & \alpha & \alpha+1 \\ \alpha & 0 & \alpha & 0 & \alpha \\ \alpha+1 & 0 & \alpha+1 & \alpha & 1 \\ \end{array}
I am not these two tables right or not.
For irreducible polynomial at degrees $1$, I think they are $\{x,x+1,x+\alpha,x+\alpha+1\}$
Can someone check those tables and give me a hint to find the remaining polynomials? Thanks