Show that the field $L = \mathbb F_3[X]/\langle X^5 - X + 1\rangle$ consists of $243$ elements and that $[X][f] = [1]$ for some $[f] \in L$.
I know that $L$ is a ring, so it is also a group with respect to $+$. However I how no idea how to calculate the size of $L$ or finding $[f]$ such that $[X][f] = [1]$ other than by brute-force, which is a terrible approach to solving a problem of this size.
Also I've proven that $\langle X^5 - X + 1\rangle$ is a maximal ideal and that $X^5 - X + 1$ is irreducible.
Could someone give me a hint or help to fit the "loose ends" here ?