1.34. This exercise begins the study of squares and square roots modulo p. (a) Let p be an odd prime number and let b be an integer with pb. Prove that either b has two square roots modulo p or else b has no square roots modulo p. In other words, prove that the congruence X² = b (mod p) has either two solutions or no solutions in Z/pZ. (What happens for p = 2? What happens if p | b?) (b) For each of the following values of p and b, find all of the square roots of b modulo p. (i) (p, b) = (7,2) (iii) (p, b) = (11,7) (ii) (iv) (p, b) = (11,5) (p, b) = (37,3) (c) How many square roots does 29 have modulo 35? Why doesn't this contradict the assertion in (a)? (d) Let p be an odd prime and let g be a primitive root modulo p. Then any number a is equal to some power of g modulo p, say a = gk (mod p). Prove that a has a square root modulo p if and only if k is even.