If n is composite then there exists a,b both less than n such that n = ab. Thus

To give a concrete example,

To prove is a field whenever n is prime, one really needs to worry only about checking whether every non zero element has a multiplicative inverse. The rest of the axioms are easy to verify.

We claim that if must be a permutation of .

Observe that if two elements, say ia and ja, in S are equal then n|(i-j)a. But n is prime, which means n|a or n|(i-j). if i is not equal to j, both are positive and less than n. Thus both the conditions are impossible. Hence i=j and this establishes all the elements of S are unique.

Now observe that and S has n distinct elements. Thus there must exist a "ja" in S such that . That