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(Solved):   6. (1) Let \( a \) be a primitive root of odd prime \( p \) with \( (a, p)=1 \). Prove tha ...



6. (1) Let \( a \) be a primitive root of odd prime \( p \) with \( (a, p)=1 \). Prove that \( a^{\frac{p-1}{2}} \equiv-1(\bm

 

6. (1) Let \( a \) be a primitive root of odd prime \( p \) with \( (a, p)=1 \). Prove that \( a^{\frac{p-1}{2}} \equiv-1(\bmod p) \). (2) Let \( p \) be odd prime and \( g \) is primitive root of \( p^{2} \). Suppose \( g^{p^{k-2}(p-1)} \) is not congruent to \( 1\left(\bmod p^{k}\right) \), where \( k \) is positive integer greater than 2 . (i) Prove \( g^{p^{k-1}(p-1)} \) is congruent to \( 1\left(\bmod p^{k}\right) \), let \( t=\operatorname{ord}_{p^{k}} g \). (ii) Prove \( g \) is also primitive root of \( p^{k} \). (3) Prove that there are no primitive roots of \( 2^{k}(k>2) \).


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6) 3) A primitive root of a prime power p^k is a number g such that the powers of g modulo p^k are the non-zero residues modulo p^k. In other words, t
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