Question on modules of finite length

Question

Let $K$ be a field and let $f$ be a nonzero polynomial on $K$. Then $K[X]/\langle f\rangle$ is a both a $K$-module and a $K[X]$-module. It can be shown that $\ell_{K[X]}(K[X]/\langle f\rangle)\leq \ell_{K}(K[X]/\langle f\rangle)$. But how to show that $\ell_{K}(K[X]/\langle f\rangle)=\deg f$? Any suggestion is appreciated.

Answer 1

The length as $K$-module is equal to the dimension as $K$-vector space. Therefore, it is enough to note that $\{1,X, \dots, X^{d-1}\}$ is a basis for the $K$-vector space $K[X]/\langle f \rangle$.