What is the intuition behind the law of quadratic reciprocity?

Question

Law of quadratic reciprocity states as follows:

Law of quadratic reciprocity — Let $p$ and $q$ be distinct odd prime numbers, and define the Legendre symbol as:

$$ \left(\frac {q}{p}\right)=\left\{\begin{array}{rl} 1 & \text{if } n^2\equiv q \pmod p \text{ for some integer } n, \\ -1 & \text{otherwise.} \end{array} \right.$$

Then:

$${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}$$

---

I have heard from class that there are hundreds of proof of this theorem. But the proof that I have learned in class is a very elementary one. As we know that many theorems in number theory have some very nice explanations using abstract algebra. Is there a proof of this theorem from the perspective of abstract algebra? And what is the intuition behind it? Thank you!

Answer 1

It would help if you told us what proof or proofs you already know.

From the perspective of algebraic number theory, the quadratic reciprocity law can be described using the cyclotomic field $\mathbf Q(\zeta_p)$ and its unique quadratic subfield, which is $\mathbf Q(\sqrt{p^*})$ for $p^* = (-1)^{(p-1)/2}p$, where $p$ is an odd prime.

Write the quadratic reciprocity law (for two different odd primes $p$ and $q$) as $$ \left(\frac{q}{p}\right) = \left(\frac{p^*}{q}\right). $$ The intuition behind this formula is that, using the field extension $\mathbf Q(\zeta_p) \supset \mathbf Q(\sqrt{p^*})$ mentioned above, the two sides of this equation describe in different ways the Frobenius element associated to $q$ in ${\rm Gal}(\mathbf Q(\sqrt{p^*})/\mathbf Q)$, which as a quotient ${\rm Gal}(\mathbf Q(\zeta_p)/\mathbf Q) \cong (\mathbf Z/(p))^\times$ is $(\mathbf Z/(p))^\times$ modulo its squares. For further details about this approach you need to learn algebraic number theory; this proof can't be described at an intuitive level. That's part of what makes the quadratic reciprocity law so mysterious.

Answer 2

The proof that best fits the intuition represented by this formula is Eisenstein's one.

Briefly, Eisenstein uses Gauss' lemma and finds that $(\frac{p}{q})(\frac{q}{p})$ equals $-1$ or $+1$ depending on the parity (odd/even) of the number of lattice points in the square with as boundary lines the $x$-axis, the $y$-axis, and the lines $x=\frac{p}{2}$ and $y=\frac{q}{2}$.

And that number is clearly equal to $\frac{p-1}{2}\cdot\frac{q-1}{2}$