ProveFor any integer p with p1 p is prime iff for all rs tha

Prove:For any integer p with p>1,

p is prime iff for all r,s that is an integer,if r>1 and s>1, then rs is not equal to p.

Thanks.

Solution

. If p is composite, say p = rs with 1 < r < p , 1 < s < p then rs is 0 in Z_p but r and s are nonzero, so Z_p is not a field.

If p is prime and a is not zero in Z_p then the greatest common divisor of a and p is 1, and consequently there are integers x and y such that ax + py = 1.

In Z_p this becomes ax = 1, so that every nonzero element in Z_p has an inverse in Z_p, proving that Z_p is a field.