(Reflective)
The proposition that x^2 is congruent to a (mod p) has a solution if and only if a^(p-1)/2 is congruent to 1 (mod p) made me think more about primitive roots. Does this proposition imply that no primitive root (mod p) is a square of another number? The primitve root (or multiplicative generator) forms a cyclic group in group theory, so is there a definitive relation between square roots mod p, and cyclic groups?
(Difficult)
The law of quadratic reciprocity was the hardest to follow in this section. It seems we are reducing two values (a and p) by using properties which define the Jacobi symbols down to simpler terms until we can finally determine a value (+/- 1) for the Jacobi symbol. The trouble I had was understanding if there is a specific pattern to follow, if the order of reduction mattered, or if the application of each property could be done in a different order and yield the same results.
It does imply that no primitive root is a square mod p. Since the multiplicative group of the integers mod p is cyclic, any squares must be even powers of a generator and can't be generators themselves.
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