Prove that every finite integral domain is a field.

Let $F$ be a field and $f$ an irreducible polynomial in the polynomial ring $F[X]$. Prove that $F[X] /(f)$ is a field, where $(f)$ denotes the ideal generated by $f$.

Hence construct a field of 4 elements, and write down its multiplication table.

Construct a field of order 9 .