If $K$ is a commutative ring which is a finite dimensional vector space over $\mathbb C$ what can we say about the maximal ideals of $K$? What can we say if instead of $\mathbb C$ we have some arbitrary field?
If $x\in K$ then $1,x,x^2,...,x^k$ must be linearly dependent for some $k \in \mathbb N$. So $x$ satisfies a polynomial over $\Bbb C$, say $f$. How is $\Bbb C[X]/\langle f\rangle$ related to $K$ if $f$ is the minimal degree polynomial?