if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
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By the irreducibility of $f(x)$, the quotient $F[x]/(f(x))$ is a field.
Show that distinct polynomials in $F[x]$ of degree $<n$ are inequivalent modulo $(f(x))$. This will follow from the fact that the polynomial ring $F[x]$ is an Euclidean domain.
This reduces the problem to counting the number of polynomials of degree $<n$ over $F$, which is straightforward.