If you want a general definition of prime, you would say that a prime is an element p which p|a*b implies p|a or p|b, with the additional restriction that p|1 i

If you want a general definition of prime, you would say that a prime is an element p which p|a*b implies p|a or p|b, with the additional restriction that p|1 is false (such elements are called “units”). This lets you extend the notion of “prime” to rings in general, including objects like polynomials or matrixes. If you are curious about some of the details, we can add that “prime” is not quite the same as “irreducible”, although the prime elements and irreducible elements are the same in the more familiar rings. The reason why “prime” doesn’t make much sense in Galois fields is precisely because it’s a field—technically, there are no primes in Galois fields, because every nonzero element divides 1, therefore every nonzero element is a unit, therefore not a prime. So there are no primes in Galois fields. This is mentioned in the article.