I have the polynomials $p(X)=X^{5}+P_4X^{4}+P_3X^{3}+P_2X^{2}+P_1X+P_0 \in\mathbb{Z_2[x]}$
I need to determine all $p(x)$ that can be factored into irreducible polynomials of degree three and two.
The catch is that I cannot use most of the techniques usually described to solve this kind of problem. What would be the steps I need to take in order to solve this problem?
For the record I am using Gallian: contemporary abstract algebra as a book and can only use theorems 17.1 and 17.5(plus corollary 1 and 2) from that specific chapter. For reference I wrote them down below:
17.1: Let $F$ be a field. If $f(x) [ F[x]$ and deg $f(x)$ is 2 or 3, then $f(x)$ is reducible over $F$ if and only if $f(x)$ has a zero in $F$.
17.5: Let $F$ be a field and let $p(x) \in F[x]$. Then $<p(x)>$ is a maximal ideal in $F[x]$ if and only if $p(x)$ is irreducible over $F$.
Corollary 1: Let $F$ be a field and $p(x)$ be an irreducible polynomial over $F$. Then $F[x]/<p(x)>$ is a field.
Corollary 2: Let F be a field and let $p(x), a(x), b(x) \in F[x]$. If $p(x)$ is irreducible over $F$ and $p(x) | a(x)b(x)$, then $p(x) | > a(x)$ or $p(x) | b(x)$.
Use the no roots criterion to determine all irreducible quadratics and cubics. Quadratics are very simple, there is only $x^2+x+1$. Cubics are a little more work. But not much. The cubic has to have shape $x^3+ax^2+bx+1$ where there is an odd number of $1$'s. Once you have your list of cubics, multiply each by $x^2+x+1$.