How do Galois fields of sizes other than powers of primes behave?

Question

I understand how Galois fields of prime powers (e.g. $GF(2^8)$, $GF(17)$, $GF(257)$) behave. What I could never figure out is how they behave with sizes other than simple powers of primes. For example, $GF(15)$, $GF(100)$, or $GF(2^43^5)$.

Is it even possible?

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Answer 1

The order of any finite field is a prime power. Further, there is only one field (up to isomorphism) of a given finite order, if such a field exists.