Galois Field of Order q Exists iff q is Prime Power

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

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

Sufficient condition
Let $\struct {F, +, \cdot}$ be a field of order $q$.

By Characteristic of Galois Field is Prime, the characteristic of $F$ is a prime number $p$.

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

By Vector Space on 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 Same Dimensional Vector Spaces are Isomorphic, this means that with $k$ equal to the dimension of $F$ there is an isomorphism of vector spaces:
 * $F \simeq \F_p^k$

Finally by the definition of the product of cardinals:
 * $\card F = \card {\F_p}^k = p^k$

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