1. (Difficult): There's a couple things I'm not following. In the proof of the very first proposition, how do we know a primitive root exists? Is there always one for every modulus? Or just for every prime modulus?
Also, why does j(p-1)/2 congruent to 0 (mod p-1) imply j is even?
And I believe the(2nd) proposition with all the properties of the Lengendre symbol, but I don't quite follow the why of all of them. Particularly the second one.
So to recap, using the Jacobi symbol -1 means the top number is not a square mod the bottom and +1 means the top number is a square mod the bottom if the top is a square for mod each of the prime factors of the bottom number. (What if the power of these prime factors isn't one?) Otherwise, the top number still isn't a square mod the bottom number. Is that correct?
2. (Reflection): The Lengendre symbol only works for modulus p, an odd prime. Good to remember.
Practicing this will be helpful.
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