$\forall n \in \mathbb{N}: [X^n]$ is invertible in $K[X]/(X^3+X^2+1)$.

Question

Setting: Field $K$

I have to show that $\forall n \in \mathbb{N}: [X^n]$ is invertible in $K[X]/(X^3+X^2+1)$.

I have simply no idea.

Answer 1

Simple: modulo $X^3+X^2+1$, we have $$X(-X^2-X)=1,$$ which means the inverse of $X$ is $-X^2-X$. Hence all powers of $X$ are invertible modulo $X^2+X^2+1$.