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?

Searching for this yielded no results.

"Is it even possible?" No. See Order of finite fields is $p^n$ — Arnaud D., Jun 07 '17 at 09:33

score 2 · Answer 1 · answered Jun 07 '17 at 09:34

2

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.

answered Jun 07 '17 at 09:34

florence

12,819

Also existence is a duplicate, see here. – Dietrich Burde Jun 07 '17 at 09:36

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

1 Answers1