How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreductible in $\mathbb{F}_{2}$

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreducible over $\mathbb{F}_{2}$.

The main 2 "weapons" I have at my disposal is the Eisenstein criteria and reduction criteria but neither seem to work in this case as $\mathbb{F}_{2} = \left\{ 1,2\right\} $.

Solution 1:

You can observe that this is the fifth cyclotomic polynomial, ${x^5-1\over x-1}$, so any field element $\alpha$ making it $0$ is a primitive fifth root of unity. That is, $\alpha^5=1$ and no smaller power will achieve this. Finite fields $\mathbb F_q$ always have cyclic multiplicative groups $\mathbb F_q^\times$, necessarily of order $q-1$. For such a field to contain a primitive fifth root of unity is equivalent to $5|q-1$. For $q$ a power of $2$, the smallest power with this property is $2^4$. Thus, $\mathbb F_{2^4}$ is the smallest field extension of $\mathbb F_2$ containing such $\alpha$. Since this is of degree $4$ over $\mathbb F_2$, the minimal polynomial of $\alpha$ is irreducible (of degree $4$).

Solution 2:

You can simply try to factor it; if it is reducible, it must have either a linear or quadratic factor. There aren't many linear and (irreducible) quadratic polynomials in $\Bbb{F}_2[X]$.