Prove that Z_m is a field iff m is a prime.

Hi,

**problem:**

Let $\displaystyle m$ be an integer, $\displaystyle m\geq2$ and let $\displaystyle \mathbb{Z}_m$ be the set of all positive integers less than $\displaystyle m$, $\displaystyle \mathbb{Z}_m=\{0,1,\cdots,m-1\}$.

If $\displaystyle \alpha$ and $\displaystyle \beta$ are in $\displaystyle \mathbb{Z}_m$, let $\displaystyle \alpha+\beta$ be the least positive remainder obtained by dividing the (ordinary) sum of $\displaystyle \alpha$ and $\displaystyle \beta$ by $\displaystyle m$,

and similarly, let $\displaystyle \alpha\beta$ be the least positive remainder obtained by dividing the (ordinary) product of $\displaystyle \alpha$ and $\displaystyle \beta$ by $\displaystyle m$.

(a) Prove that $\displaystyle \mathbb{Z}_m$ is a field if and only if $\displaystyle m$ is a prime.

(b) What is -1 in $\displaystyle \mathbb{Z}_5$?

(c) What is $\displaystyle \frac{1}{3}$ in $\displaystyle \mathbb{Z}_7$?

**attempt:**

(a) I could really use a hint or two on this one..

(b) $\displaystyle remainder\;of\;\frac{1+(x)}{5}=0$

Answer: 4.

(c) $\displaystyle remainder\;of\;\frac{3\cdot(x)}{7}=1$

Answer:5.

Btw, is there a nice math symbol for "the remainder of"?

Thanks.