Quadratic reciprocity — flashcards
Every card the map can derive about Quadratic reciprocity — definitions, statements, attribution, and the edges it sits on.
Theorems 5
State: Law of quadratic reciprocity.
For distinct odd primes p, q the Legendre symbols satisfy (p|q)(q|p) = (−1)^((p−1)/2·(q−1)/2).
derived from st/quadratic-reciprocity by the statement rule
State: Hilbert symbol product formula.
For a, b in ℚ*, the product of the local Hilbert symbols over all places equals 1. Reciprocity is this statement, and it is why the completions are not independent.
derived from st/hilbert-product by the statement rule
State: Hasse–Minkowski theorem.
A quadratic form over ℚ has a nontrivial rational zero if and only if it has one over ℝ and over every ℚₚ. The first major local–global principle.
derived from st/hasse-minkowski by the statement rule
State: Explicit formula for the 2-adic Hilbert symbol.
For a = 2^m·u, b = 2^n·v with u, v odd 2-adic units, (a,b)₂ = (−1)^(ε(u)ε(v) + mω(v) + nω(u)), where ε(u) = (u−1)/2 mod 2 and ω(u) = (u²−1)/8 mod 2 — both fixed-depth congruences (u mod 4, u mod 8) independent of the size of u.
derived from st/hilbert-2adic-explicit by the statement rule
State: The 2-adic square-class shortcut has no odd-prime analogue.
By Hensel's lemma, whether an odd 2-adic unit u is a square depends only on u mod 8 — a fixed-width digit-read, however large u is. No analogous fixed k exists at an odd prime q: whether u is a quadratic residue mod q is not determined by any bounded number of low bits of u. Checked directly: u=3 vs u=11 (agreeing mod 8) disagree on 339/666 tested primes q; u=7 vs u=71 (agreeing mod 64) disagree on 500/1004 — matching more shared bits does not shrink the disagreement rate toward 0. This is the precise sense in which quadratic reciprocity's supplementary laws are free lookups while the reciprocity step itself is not, and it is what st/hilbert-product's 2-adic term is actually doing.
derived from st/2adic-square-shortcut-is-unique by the statement rule
Names and dates 3
Law of quadratic reciprocity — who, and when?
Carl Friedrich Gauss, 1801.
derived from st/quadratic-reciprocity by the attribution rule
Hilbert symbol product formula — who, and when?
David Hilbert, 1897.
derived from st/hilbert-product by the attribution rule
Hasse–Minkowski theorem — who, and when?
Hermann Minkowski, then Helmut Hasse, 1890 (Minkowski), 1923 (Hasse).
derived from st/hasse-minkowski by the attribution rule
Formalisation 1
What is Law of quadratic reciprocity called in Mathlib?
legendreSym.quadratic_reciprocity — Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
derived from st/quadratic-reciprocity by the binding rule
Connections 2
What is the correspondence between Quadratic reciprocity and Class field theory? [Correspondences]
Quadratic reciprocity is the first visible instance of the pattern class field theory completes for all abelian extensions.
derived from correspondences:quadratic-reciprocity→class-field-theory by the connection rule
What is the correspondence between Quadratic reciprocity and p-adic numbers? [Correspondences]
Reciprocity is the statement that the local Hilbert symbols over every completion multiply to 1. So the p-adic worlds are not independent — fixing all but one determines the last. This was your own question: whether quadratic reciprocity means the completions are coupled. It does.
derived from correspondences:quadratic-reciprocity→p-adic-numbers by the connection rule
Concepts 1
What is Quadratic reciprocity, in one sentence?
Whether p is a square mod q is determined by whether q is a square mod p. Gauss called it the golden theorem; it is the seed of the whole reciprocity story.
derived from quadratic-reciprocity by the concept rule