Galois Field of Order q Exists iff q is Prime Power

Theorem
Let $q \ge 0$ be a positive integer.

Then there exists a finite field of order $q$ iff $q$ is a prime power.

Sufficient condition
Suppose that $\left({F,+,\cdot}\right)$ is a field of order $q$.

By Finite Field has Non-Zero Characteristic, $F$ has characteristic $p > 0$.

By Characteristic of Integral Domain, $p$ is a prime number.

By Field of Prime Characteristic has Unique Prime Subfield the prime subfield of $F$ is $\F_p := \Z/p\Z$.

By Field Extension is Vector Space, $F$ is an $\F_p$-vector space.

Since $F$ is finite, $F$ has a finite basis over $\F_p$.

By Expression of Vector as Linear Combination from Basis is Unique, this means that for some $k \in \N$ there is an isomorphism of vector spaces:
 * $F \simeq \F_p^k$

Finally by the definition of the product of cardinals,
 * $\left\vert F \right\vert = \left\vert \F_p \right\vert^k = p^k$

So the order of $F$ is a prime power.