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