Ring of Integers Modulo Prime is Field/Proof 2

Proof
Let $p$ be prime.

From Irreducible Elements of Ring of Integers, we have that $p$ is irreducible in the ring of integers $\struct {\Z, +, \times}$.

From Ring of Integers is Principal Ideal Domain, $\struct {\Z, +, \times}$ is a principal ideal domain.

Thus by Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal, $\ideal p$ is a maximal ideal of $\struct {\Z, +, \times}$.

Hence by Maximal Ideal iff Quotient Ring is Field, $\Z / \ideal p$ is a field.

But $\Z / \ideal p$ is exactly $\struct {\Z_p, +, \times}$.

Let $p$ be composite.

Then $p$ is not irreducible in $\struct {\Z, +, \times}$.

Thus by Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal, $\ideal p$ is not a maximal ideal of $\struct {\Z, +, \times}$.

Hence by Maximal Ideal iff Quotient Ring is Field, $\Z / \ideal p$ is not a field.