We say that x is a square root of a in Z ? p if x^ 2 ? a mod( p ). a) Find all t

Question

We say that x is a square root of a in Z?p  if x^2 ? a mod(p).

a)  Find all the square roots of ?1 mod (13). Justify your answer.

b)  It is known that every number in Z?p has either no square roots or two square roots. If x1 and x2 are two square roots of a in Z?p, what is x1 + x2 mod (p)? Justify your answer.

c)  Let p be an odd prime number and let g be a primitive root for Z?p. Show that x^2 ? a mod (p) has solution if and only if a ? g2k mod (p), with k ? Z.

d)  Let p be an odd prime number and let b be a solution of x^2 ? a mod (p). Show that there exists k?Z such that b+kp is a solution of x^2 ?a mod(p^2).

Explanation / Answer

ALL SYMBOLS USED ARE INTEGERS 1

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