Show that there are infinitely many irreducible polynomials in Zp[x], where p is prime.
Deduce that there are infinitely many non-isomorphic finite fields of order a power of p.
Mimick Euclides' proof for primes in $\displaystyle \mathbb{N}$: suppose $\displaystyle p_1(x),\ldots,p_n(x)$ are all the irreducible polynomials in $\displaystyle \mathbb{Z}_p[x]$ ,and define $\displaystyle p(x)=p_1(x)\cdot\ldots\cdot p_n(x)+1$ ...