Ring of Integers Modulo Prime is Field/Proof 4

Proof
Let $m$ be prime.

From Ring of Integers Modulo Prime is Integral Domain, $\struct {\Z_m, +, \times}$ is an integral domain.

From Finite Integral Domain is Galois Field, $\struct {\Z_m, +, \times}$ is a field.

Now suppose $m \in \Z: m \ge 2$ is composite.

From Ring of Integers Modulo Composite is not Integral Domain, $\struct {\Z_m, +, \times}$ is not an integral domain.

From Field is Integral Domain $\struct {\Z_m, +, \times}$ is not a field.