Is $X^5+...+1 \in \mathbb{F_2}[X]$ irreducible?

Question

I am trying to determine if the following polynomials are irreducible in $\mathbb{F_2}[X]$ are irreducible:

$f(X)=X^5+X^2+1$

$g(X)=X^5+X^3+1$

There are no linear factors since $f(0)=f(1)=g(0)=g(1) \neq 0$

However there is another possible factorization: $(X^3+aX^2+bX+c)(X^2+dX+e)$

I am unsure how to verify if $f$ and $g$ can be factored in this way in $\mathbb{F_2}[X]$

Would appreciate your help on this one

Answer 1

Hint: find all irreducible polynomials of degree 2 (there are not many) and try to divide $f$ and $g$ by this (those) irreducible polynomial(s) of degree 2

Answer 2

From H. Potter's hint, it follows that $f(X)$ is reducible iff it is divisible by $X^2+X+1$. Now, in the same spirit, $g(X)$ is reducible iff it is divisible by $X^2+X+1$ as well. As $f(X)$ is reducible iff $g(X)$ is reducible, and $f(X)+g(X)=X^3+X^2=X^2(X+1)$ is not divisible by $X^2+X+1$, we conclude that $f(X)$ and $g(X)$ are irreducible over $\mathbb{F}_2$.