find all composite order fields between 200 and 900 .

Question

There are many fields of composite order between 200 and 900 how can I find those fields.

Answer 1

All finite fields have order $p^n$ for a prime $p$ and a positive integer $n$; of these, the ones with composite order are those where $n > 1$. So you need to find all choices of $p, n$ so that $200 \le p^n \le 900$ and $n > 1$. For example, the only powers of $2$ that work are $2^8 = 256$ and $2^9 = 512$. GIven that $\sqrt{900} = 30$, you can stop at $p = 29$.