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,

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

comment SXK66U5CG8H78ZD3

Surely a clearer way to state this is:

(p/q)=-(q/p) only if both p,q==3 (mod 4)
Otherwise, (p/q)=(q/p) (where p,q, distinct odd primes).

Comment by jr512 Thu 02 Jun 2011 14:44:53 UTC