Are the Integers modulo 24 not a field or an integral domain?

Question

As the order of the group is not prime, am I right to assume that the group can not be a field or an integral domain?

Answer 1

Not exactly. Every finite field has order equal to some power of its characteristic (namely, the additive order of $1$). Since the characteristic in an integral domain must be prime (for otherwise, the product of two nonzero elements would be zero), it follows that every finite field has prime power order. In fact, there exists a unique finite field of order $p^k$ for every prime $p$ and positive integer $k$ (it is simply the splitting field of the polynomial $x^{p^k} - x$ over $\mathbb{Z}/p\mathbb{Z}$).

The reason that $\mathbb{Z}/24\mathbb{Z}$ is not an integral domain (with respect to the multiplication law induced from the integers) is that $3 \cdot 8 = 24 \equiv 0$, but neither $3$ nor $8$ is congruent to zero.