I've been solving problems from my Galois Theory course, and I don't find the way to solve this one:
Given the polynomial $f(X)=X^4+X+1\in\mathbb F_2[X]$, prove that $f(X)$ is irreducible in $\mathbb F_2[X]$. If $a$ is a root of $f(X)$, prove that $\mathbb F_2(a)$ is a splitting field of $f(X)$ over $\mathbb F_2$.
The work I've done so far:
Firstly, to prove $f(X)=X^4+X+1$ is irreducible in $\mathbb F_2[X]$, I calculated all degree 4 non-irreducible polynomials (I did it by calculating all products possible of the degree 2 polyniomials, and $f(X)$ was not one of them. I just need to check this because, since $f(X)$ does not have roots in $\mathbb{F_2}$, the only way it can be reduced is by being product of two polynomials of degree 2).
After proving $f(X)$ is irreducible, I assumed $a$ is one of its roots, and here is where I don't know how to continue. I know that in order for it to be splitting field, it must have all roots of $f(X)$, so I must prove that, being $\beta, \gamma, \delta$ the other roots of $f(X)$, they verify that ther are inside $\mathbb F_2(a)$.
I tried Ruffini and got that $$f(X)= (X-a)(X^3+aX^2+a^2X+1+a^3)$$ but I don't know how to continue. I considered using Vieta's formulas, but I'm not sure if that will help and how to properly use them in this case.
Is what I have already done correct? How can I end this problem? Any help or hint will be appreciated, thanks in advance.
Your approach to show irreducibility is correct. Let me give a hint for the second part of the question.
You are working over the finite field $\mathbf{F}_2$, so you have Frobenius ($x\mapsto x^2$) at your disposal. Let $\alpha\in \overline{\mathbf{F}}_2$ be a root of $f=X^4+X+1\in \mathbf{F}_2[X]$. What is the orbit of $\alpha$ under Frobenius ? How can this be used to calculate the other roots of $f$ (and even prove irreducibility of $f$) ?