Finite field with prime numbers

[Glass]

Homework Statement

If F is a finite field show that there is a prime p s.t. (p times)a+a+...+a=0 for all a in the field

Homework Equations

The Attempt at a Solution

Well I managed to prove that there must be an a in F s.t. (prime number, call p, times)a+a+...+a=0 but I can't seem to prove that for every a in F (p times)a+a+...+a=0 (This is the only approach I could think of).

 

[morphism]

If you showed that pa=0 for some a, then pb = (pa)(a^-1 b) = 0 as well.

Another approach goes as follows. Define a homomorphism f from Z (the ring of integers) into F by n -> n*1. Now Z/ker(f) is a subring of F, hence an integral domain, and consequently ker(f) is a prime ideal of Z. If F is finite, f cannot be an injection, so ker(f) isn't trivial, and is thus of the form pZ for some prime p. This means Z/pZ sits inside F, and in particular p=0 in F.