Proving that if the cardinality of a field $F$ is finite and equal to $q$, then the ring $F[X]/(X^n)$ is finite of cardinality $q^n$

Question

I'm trying to prove how the cardinality of a field $F$ is finite and equal to $q$, then the quotient ring $F[X]/(X^n)$ is finite of cardinality $q^n$.

How do I go about this when the quotient ring is not an integral domain?

Thank you!

Answer 1

This essentially amounts to counting the cosets of $F[x]/(x^n)$. The set of elements $\{a_0+a_1x+\cdots +a_{n-1}x^{n-1}\mid a_i\in F \}$ is a complete set of representatives for the cosets. This set "clearly" has $q^n$ elements if $q=|F|$. Can you now justify why this is a complete set of representatives?