Finite field with nonprime cardinality In my textbook for discrete mathematics the following is stated:

Theorem: $\mathbb Z_p$ is a field if and only if $p$ is prime.
In the following we denote the field with $p$ elements by $GF(p)$ rather than $Z_p$. As explained later, "$GF$" stands for Galois field. Galois discovered finite fields around 1830.

However, there is a field with $p=4$ elements (right?) and clearly $4$ is not a prime. I  think that I am misunderstanding something fundamentally. Is it maybe that there are fields other than $\mathbb Z_p$ (what is the spoken name of this set?) that do not need to be of prime cardinality?

Yes, there are finite fields other than $\Bbb Z_p$. The cardinal of such a field is always the power of a prime number. And, yes, there is a field with $4$ elements. It can be defined as $\Bbb Z_2[x]/\langle x^2+x+1\rangle$.
