Let gcd(a, p) = 1 with p a prime. Show that if a has at...
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Let gcd(a, p) = 1 with p a prime. Show that if a has at leastone square root, then a has exactly 2 roots. [hint: look atgenerators or use x^2 = y^2 (mod p) and use the fact that ab = 0(mod p) the one of a or b must be 0(why?) ]
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Solution 1 Suppose that y is a square root of a mod p Thus the quadratic congruence x2 a mod p has a solution y mod p Therefore x2 y2 mod p and hence x2y2 0 mod p Hence p x2 y2 and thus p xyxy Since p is a prime p xy or p
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