Finite field with nonprime cardinality

Question

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?

Answer 1

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 $\Bbb Z_2[x]/\langle x^2+x+1\rangle$.

Answer 2

Any finite field must be order of $p^{n}$ , where $p$ prime and $n$ is some non negative integer.

For order $p^{n}$ , just take Algebraic extension of the field $\mathbb{Z}_{p}$ is of degree $n$, So, field with order $p^{n}$ can be represented as the quotient field $\frac{\mathbb{Z}_{p}[x]}{\langle\, p(x)\,\rangle} $, where $p(x)$ is some irreducible polynomial over $\mathbb{Z}_{p}$.