Ring of Integers Modulo Prime is Field

Theorem
Let $m \in \Z: m \ge 2$.

Let $\struct {\Z_m, +, \times}$‎ be the ring of integers modulo $m$.

Then:
 * $m$ is prime


 * $\struct {\Z_m, +, \times}$ is a field.
 * $\struct {\Z_m, +, \times}$ is a field.

Prime Modulus
$\struct {\Z_m, +, \times}$‎ is a commutative ring with unity by definition.

From Reduced Residue System under Multiplication forms Abelian Group, $\struct {\Z'_m, \times}$ is an abelian group.

$\Z'_m$ consists of all the elements of $\Z_m$ coprime to $m$.

Now when $m$ is prime, we have, from Reduced Residue System Modulo Prime:
 * $\Z'_m = \set {\eqclass 1 m, \eqclass 2 m, \ldots, \eqclass {m - 1} m}$

That is:
 * $\Z'_m = \Z_m \setminus \set {\eqclass 0 m}$

where $\setminus$ denotes set difference.

Hence the result.

Composite Modulus
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.

We have that a field is an integral domain.

Therefore $\struct {\Z_m, +, \times}$ is not a field.

Also see

 * Definition:Field of Integers Modulo Prime
 * Galois Field