Homomorphic images of $\mathbb{Z}[x]$

Question

How to prove that any finite field is a quotient ring of $\mathbb{Z}[x]$ ?

I am not sure whether this result is true or false.

Any hint will be appreciated.

Thanks in Advance.

Answer 1

Of course, there is a unique field (up to isomorphism) for every prime $p$ and positive integer $n$ such that the field has order $p^n$.

If your finite field has characteristic $p>0$, you just need to find a candidate for a quotient of $\mathbb Z_p[x]$ which has $p^n$ elements.

All you have to do is prove that there is an irreducible polynomial of degree $n$. This is actually pretty simple to do via a counting argument. The number of monic polynomials of degree $n$ is $p^n$, and one can show inductively that there is always a deficit between this number and the number of composite polynomials of the same degree.

Once you find your irreducible polynomial $f(x)\in F_p[x]$ of degree $n$, you have that $\mathbb Z[x]/(p, f(x))\cong F_{p^n}$.