Lecturer and textbooks tell me that this theorem is certainly the most important theorem in Number Theory and (therefore?) also probably the most important theorem in all of Mathematics. There are hundreds of proofs; first one was by Gauss at the age of 19. Lecturer: “if Number Theory is the queen of Mathematics, then this theorem is its jewel.”

I do not yet fully grasp what the theorem even says (because I haven’t looked at the proof properly) and I certainly don’t see why it’s so important yet, but I’m posting it here because I rather like the idea that the deepest stuff in Maths lies in ℤ.

Definition. Let p > 2 be prime and let a ∈ ℤ be s.t. $p \nmid a$; a is a unit mod p. Then a is a quadratic residue of p if ∃x ∈ ℤ s.t. x² ≡ a (mod p) and a is a quadratic non-residue if not.

Notation. For any a ∈ ℤ we define the Legendre symbol: $$ \left(\frac{a}{p}\right) \mathrel{\mathop:}= \left\{\begin{array}{cl} +1 &p\nmid a \text{ and } a \text{ is a QR of } p, \ -1 &p\nmid a \text{ and } a \text{ is a QNR of } p, \ 0 &p|a.\end{array}\right. $$

Theorem. (The Law of Quadratic Reciprocity, Gauss, 1796) For p, q distinct odd primes,

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