Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$.
Any finite field $F$ is of the order $p^n$ where $p$ is a prime and $n\in \Bbb N$ .
If we want to make a field of order say $p^n$ then take the quotient ring $\Bbb Z[x]/\langle p \rangle$ and then form the field $\Bbb Z_p[x]$ .
Then choose an irreducible polynomial of degree $n$ in $\Bbb Z_p[x]$ and form the quotient $\Bbb Z_p[x]/\langle f(x)\rangle $ and we get a field of order $p^n$ elements.
Is the result correct?
Is my proof correct?