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Connection between Gauss's lemma and Zolotarev's lemma

So I was reflecting on the relationship between Gauss's Lemma and Zolotarev's Lemma in proofs of quadratic reciprocity: GL: $(a/p) = -1^n$, where $n$ is the number of least positive residues of $ax$ ...
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Density of “simultaneous squares”

Let $(u,v)$ be a pair of non-zero integers. We say that $(u,v)$ is a pair of simultaneous squares if for all primes $p$ dividing $u$, we have $\left(\frac{v}{p}\right) = 1$ and for all primes $q$ ...
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nonabelian reciprocity law

I heard the following relation in a talk by Peter Scholze. Could someone explain "in a simple way" what is the precise relation between the polynomial $x^4-7x^2-3x+1$ and the integral homology of the ...
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Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
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Given n and q, how to find p so q$\neq$n-th power (mod p)?

Reasonable exceptions allowed on $q$. Example solution: $n=2$. Suppose $q$ is odd. Let $p$ be so $pq\equiv -1$ (mod 8). Then $q\neq$ 2nd power (mod $p$) is the same as $\left(\frac{q}{p}\right)=-1$...
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Computation of Hilbert symbol of order 4

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for the ...
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Can Eisenstein's lattice point proof of quadratic reciprocity be generalized?

I'm referring to this proof. The key formula ("Eisenstein's Lemma") is $$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where u=2,4,\ldots,p-1}$$ The sum in the exponent ...