All irreducible polynoms of degree 4 in $\mathbb{Z}_2$ I need to find all irreducible polynoms of degree at most 4 in $\mathbb{Z}_2$
This is my result:
$$ x^4 + x^1 + 1 \\
x^4 + x^2 + 1\\
x^4 + x^3 + 1  \\
x^4 + x^3 + x^2 + x^1 + 1 \\
$$
Is this correct?

$x^4 + x^2 + 1= (x^2 + x + 1)^2$ is not irreducible.
An irreducible polynomial of degree $4$ defines a field extension of $\mathbb Z_2$ of degree $4$. Its $2^4$ elements are the roots of $x^{2^4}-x$ because of Lagrange's theorem in group theory.
Therefore, the irreducible polynomials of degree $4$ are those that divide $x^{2^4}-x$ over $\mathbb Z_2$:
$$
x^{16}-x=x (x + 1) (x^2 + x + 1) (x^4 + x + 1) (x^4 + x^3 + 1) (x^4 + x^3 + x^2 + x + 1)
$$
and so there are exactly three of them.
To factor $x^{16}-x$, factor $x^{15}-1$, starting with the factorization into cyclotomic polynomials:
$$
x^{16}-x = x(x^{15}-1)=x \Phi_1(x) \Phi_3(x) \Phi_5(x) \Phi_{15}(x) 
$$
which gives
$$
x^{16}-x = x(x - 1)(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1) (x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)
$$
We then need to further factor these mod 2. The last factor is the only one that is reducible.
