Find all the monic irreducible polynomials of degree less than equal to $3$ in $F_2[X]$, and the same in $F_3[x]$.
We need to have $1$ in the expression of our polynomial since otherwise our polynomial will have $0$ as a root and will therefore not be irreducible. So starting with $1$ we can take $x,x^2$ and $x^3$ as the following expression. We can't stop here since the polynomials $1+x,1+x^2$ and $1+x^3$ will have $1$ as root or in other words polynomials of "length" $4$ is reducible. So we carry on: we can now add $x^2$ or $x^3$ to $1+x$ and similarly, we can add $x^3$ to the polynomial $1+x^2.$ Thus we get all the irreducible polynomials $$1+x,1+x+x^2,1+x+x^3,1+x^2+x^3.$$ Note we do not extend the polynomial $1+x^3$ since it would repeat the above cases.
For $F_3[x]$ we have in the first step $$1+x,1+x^2,1+x^3.$$ Note that $1+x^3$ is reducible. In the second step $$1+x+x^2,1+x+x^3,1+x^2+x^3.$$ If we add one more term then each polynomial will be divisible by $x+1.$ So 4 terms are out. Three terms are also out since $x+2$ divides all of them. So the final list of polynomials should be $$x+1,x^2+1.$$
Is this correct?