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Proof of two Legendre Symbol identities

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1) Use that (Z/pZ)* is cyclic to directly prove that (-3/p) = 1 when p = 1 (mod 3).

2) If p = 1 (mod 5), directly prove that (5/p) = 1.

I know that with exercise 1, we show that there is an element c in (Z/pZ)* of order 3, and that (2c+1) ^2 = -3.

Similarly for exercise 2, there is a c in (Z/pZ)* of order 5, and [(c + c^4) ^2] + (c + c^4) - 1 = 0.

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Solution Summary

In this solution, we provide proof for two algebraic equations.

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(1) First, we note that |(Z/pZ)*| = p - 1. From the condition, p = 1 (mod 3), then p - 1 = 3k for some positive integer k. Thus |(Z/pZ)*| = 3k.
Second, since (Z/pZ)* is a cyclic group, then (Z/pZ)* = <x> and x^(3k) = 1. We set c = x^k, then c^3 = x^(3k) = 1 and thus the ...

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