1. ## Rings3

Show that Zn is a field if and only if n is a prime.

2. Originally Posted by mpryal
Show that Zn is a field if and only if n is a prime.

do you know what a field is? go through and prove that $\mathbb{Z}_n$ when $n$ is prime fulfills all the conditions. note that if n is not prime you will have zero divisors (you must show this), which of course, is bad, and would make the group not a field.
If $n=ab$ where $0 then $[a]_n[b]_n = [0]_n$ but $[a]_n,[b]_n\not = 0$ so it is not an integral domain.
If $n$ is prime then for any $x\not \equiv 0(\bmod n)$ there is $y$ with $xy\equiv 1(\bmod n)$ this means $[y]_n$ is inverse of $[x]_n$.