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# Legendre symbols and primitive roots? watch

1. Let be an odd prime.

A primitive root mod is an integer with the order of

I have shown

(This is the Legendre symbol)

Here is the question:

Using primitive roots show there are the same number of quadratic residues as there are quadratic nonresidues modulo

I am aware of the "standard" proof of this result which uses Lagrange's theorem (for polynomials) but I don't know how to prove it using primitive roots.

Obviously since then any primitive root is a quadratic non residue but this is as far as I have got.

I tried to use Euler's criterion somehow but couldn't see a way forward.
2. I would think you're supposed to do something very "low level", based on the fact that every element of Z_p* can be written as for some integer k, and then the quadratic residue status can be found based on whether k is odd or even.

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Updated: January 12, 2017
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