Answer to Question #351027 in Abstract Algebra for Fed

Question

3.8 D is an integral domain and D is of finite characteristic,

prove that the characteristic of D is a prime number.

Answer 1

Assume D is the characteristic of D. Let a be a non zero element of D. Seeking a contradiction assume D is not prime. Then D can be written as a factor: rs=D for some r and some s. By definition Da=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 D is a prime.