Let D be an integral domain of finite characteristic p say P

Let D be an integral domain of finite characteristic, p, say. Prove that p is a prime integer.

Solution

Let D be an integral domain of finite characteristic p,say ,prove that p is a prime integer.

Assume p is the characteristic of D. Let a be a non zero element of D.

Seeking a contradiction assume p is not prime. Then p can be written as a factor:

rs=p      for some r and some s. By definition pa=0, so (rs)a=0.

We know that r, s are non-zero, so by definition of integral domain the only way this equation can equal zero is if a=0 ,however this is a contradiction as we chose a to be a non-zero element of D.

Therefore p is a prime.