Classify finitely-generated modules over $\mathbb{F}_2[x]/\langle x^2+x+1 \rangle$ up to isomorphism.

Question

While studying the classification of finitely-generated modules over PIDs, I came across this exercise: Classify finitely-generated modules over $\mathbb{F}_2[x]/\langle x^2+x+1 \rangle$ up to isomorphism. It would be nice to have a solution as an example to illustrate the general theory.

Thanks in advance for any help.

Answer 1

Hint:

1. Check that $ x^2+x+1 $ is irreducible in $ \mathbb F_2[x] $;
2. Prove that $ \mathbb F_2[x]/(x^2+x+1) $ is a field;
3. So what is a finitely-generated module over a field?