For any integer n, Z_n^* (Z with n-subscript and *-superscript) is
defined as 

	Z_n^* = { i in Z_n : such that i has a multiplicative inverse
in Z_n }

and it so happens that

	      = { i in Z_n : such that gcd(i,n)=1}.

This fact follows from the extended Euclidean algorithm.

Now, when n is prime then Z_n^*={1,2,...,n-1}=Z_n-{0}, i.e., all the
elements of Z_n minus zero.  That is why, when n is prime, (Z_n,+,*)
is a field.