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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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    I have approached by considering F[x] as Euclidean Ring. Also I have considered the Field formed by Quotient group of2012-11-17

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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 $ 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 $ over $F$, which is straightforward.