Let p be a prime integer.

1. Prove that Z_p is an integral domain.

2. Prove that Z_p[x] is an integral domain.

I know that if a ring is a ED it is a PID and if it is a PID it is a UFD and if it is a UFD it is an integral domain. So this can be proven by proving ED, PID or UFD. However, I'm stuck. I need a clear explanation/proof.