Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

Question

I'm not sure where to go with this question.

I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.

Answer 1

Every polynomial in $K[x]$ has a unique representative mod $p(x)$ of degree less than $d$.

This means that $L$ has dimension $d$ as a vector space over $K$.

Answer 2

You've proven that $L$ is a field. Now, let's give an explicit basis for $L/K$. If you consider the classes of $\overline{1}$, $\overline{x} \cdots \overline{x^{d-1}} $ i, you can show it's a basis of L/K. (In this case, irreducibility is used only for the fact that L is a field, but the basis stuff works for any polynomial)