# Example of field $K$ s.t. $|K|=p^{n}$ with $n\geq 2$.

Do you have a example of field $K$ s.t. $|K|=p^n$ with $n\geq 2$ ? I know that $\mathbb Z/n\mathbb Z$ is a field iff $n$ is prime, but since $p^n$ is not prime, what kind of field can be such that $|K|=p^n$. Would $$\mathbb Z/p\mathbb Z\times \mathbb Z/p\mathbb Z$$ a field ? If yes, does a field $K$ s.t. $|K|=p^n$ is necessarily isomorphic to $$\mathbb Z/p\mathbb Z\times ...\times \mathbb Z/p\mathbb Z\ \ ?$$

The way you generate a field of order $p^n$ is to find an polynomial of degree $n$ that is irreducable over $\mathbb Z / p\mathbb Z [x]$, then mod out by the ideal generated by it. This is a maximal ideal, so modding out gives you a field.
The answer is no. In $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$, we have $$(0,1)\times (1,0)=(0,0)$$ and a field cannot have zero divisors.