Integral Domain of Prime Order is Field

Theorem
The following statements are equivalent:
 * 1) $$p$$ is a prime.
 * 2) $$\mathbb{Z}_p$$ is an integral domain.
 * 3) $$\mathbb{Z}_p$$ is a field.

Proof

 * By Principal Ideal of Prime is Maximal and Maximal Ideal iff Quotient Ring is Field, (1) implies (3), and from Field is Integral Domain, (3) implies (2).


 * By the alternative definition of Integral Domain, $$\mathbb{Z}_p$$ is an integral domain iff $$\left({\mathbb{Z}_p^*, \times}\right)$$ is a semigroup.

Suppose $$p = m n$$ where $$1 < m < p, 1 < n < p$$.

Then in the ring $$\mathbb{Z}_p$$ we have $$q_p \left({m}\right) \ne 0, q_p \left({n}\right) \ne 0$$.

But $$q_p \left({m}\right) q_p \left({n}\right) = q_p \left({m n}\right) = q_p \left({p}\right) = 0$$.

Thus if $$p = m n$$, $$\left({\mathbb{Z}_p^*, \times}\right)$$ is not a semigroup.