Cryptography: Why is Zn not a Field?

•The main reason for why, in general, Zn is only a commutative ring and not a finite field is because not every element in Zn is guaranteed to have a multiplicative inverse.

•In particular, as shown before, an element a of Zn does not have a multiplicative inverse if a is not relatively prime to the modulus n.

•What if we choose the modulus n to be a prime number?

–A prime number has only two divisors, one and itself.