Question

for a prime number p the ring of integers modulo P i. e. Zp is? ​

Answer 1

Step-by-step explanation:

Suppose ab=0 for b≠0. Then ab≡0modp and so p divides the product ab. Since p is a prime, it follows that p divides a factor. So p∣a or p∣b. But b≠0 and so p∣a, i.e., a≡0modp, i.e., a=0 in Zp.

Now, for the "hence (Z/pZ)∗ = {1,2,...,p−1}" part, I know that Z/pZ = {0,1,2,...,p−1} and I already proved in a previous part of this problem that an element in a ring is either a zero-divisor or a unit. And thus, since 0 is the zero-divisor, then all the other elements of Z/pZ are units, therefore (Z/pZ)∗ = {1,2,...,p−1}. Is this correct?

For the "Z/pZ is field" part, I know that a field is a commutative ring, so all I have to do here is to show that Z/pZ is commutative