Hello littlebbygurl, I have the proof of a) for you:
If 1 has infinite order under addition, then the characteristic of the integral domain is 0.
Suppose that 1 has order n, and , where . Then, we have:
Therefore, or . However, since n is the least positive integer s.t. , it must be so that or , proving that n is prime.
If (*) is not trivial to you, you may want to prove it as well. The same is true of line one of the proof.
Hope this helps, happy mathing!