Is there a field with $n$ elements for all $n \in \mathbb{N}$?

Question

I don't think this is true, but I'm not sure. I certainly know of finite fields with 2,4 and 8 elements, and of course $p^n$ elements where $p$ is prime, for all $n \in \mathbb{N}$.

Answer 1

Let $F$ be a field of order $n$ and $P$ its prime subfield. Then $P\cong \mathbb Z_p$ where $p = \lvert P \rvert$. Thus $p$ is prime. So now $F$ is a finite $P$ vector space, and thus $n = \lvert F \rvert = \lvert P \rvert ^k = p^k$, where $k = \operatorname{dim}_P (F)$.