1. ## Question on Fields

Show that q is a prime number iff Z/qZ is a field.

Much appreciated!

2. Originally Posted by h2osprey
Show that q is a prime number iff Z/qZ is a field.

Much appreciated!
If $q$ is prime then the equation $ax\equiv 1(\bmod q)$ is always solvable for $a\not \equiv 0(\bmod q)$ and so it is a field. Conversely, if $\mathbb{Z}/q\mathbb{Z}$ and suppose $q$ is not prime, then $q = q_1q_2$ in a non-trivial factorization. But then $[q_1]_q[q_2]_q = [0]_q$, and that is a contradiction.