How do I show that if p > 0 is the characteristic of the finite field F (p is prime obviously) that F contains a subfield {k * 1F: k=0,1,...,p-1} that is isomorphic to Zp?

Note k*1F= 1+1+1...+1 (k times)

Thank you in advance.

Also, how can I show that the set {k * 1F: k=0,1,...,p-1} actually is a subfield?