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Quadratic reciprocity

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A page from Gauss's Disquisitiones Arithmeticae showing mathematical formulas about quadratic reciprocity.

In number theory, the law of quadratic reciprocity is a special rule that helps us solve quadratic equations with prime numbers. It tells us if a number can be a perfect square when we only look at the remainder after dividing by another number. This rule makes solving these problems much easier.

Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801.

The law was first stated by mathematicians Leonhard Euler and Adrien-Marie Legendre. Later, Carl Friedrich Gauss proved it and called it very important. Gauss was so excited that he found many different ways to prove it.

Quadratic reciprocity is very useful in mathematics. It helped shape modern algebra, algebraic geometry, and even led to ideas like class field theory and the Langlands program. Many mathematicians have found proofs for this theorem, showing how important it is.

Motivating examples

Quadratic reciprocity helps us find patterns in numbers that are perfect squares when we look at them in a special way.

Consider the expression ( n^2 - 5 ). When we look at the prime factors of this for different values of ( n ), we see some interesting things. Some primes, like those ending in 1 or 9, appear. Others, like those ending in 3 or 7, do not.

Quadratic residues are numbers that can be written as a square in this special way. For example, modulo 5, the numbers 1 and 4 are quadratic residues because they are squares of 1 and 2. Studying these residues shows useful patterns in number theory. For instance, the product of two quadratic residues is also a residue. These patterns help mathematicians solve tricky problems with primes and squares.

n⁠ f ( n ) {\displaystyle f(n)} ⁠       ⁠ f ( n ) {\displaystyle f(n)} ⁠       n
1−4−22162512513195622⋅239
2−1−11728422⋅713210191019
34221831911⋅2933108422⋅271
411111935622⋅893411511151
52022⋅5203955⋅7935122022⋅5⋅61
631312143622⋅1093612911291
74422⋅112247947937136422⋅11⋅31
859592352422⋅1313814391439
97622⋅192457157139151622⋅379
10955⋅192562022⋅5⋅314015955⋅11⋅29
1111622⋅292667111⋅6141167622⋅419
121391392772422⋅1814217591759
1316422⋅412877919⋅4143184422⋅461
141911912983622⋅11⋅194419311931
1522022⋅5⋅11308955⋅17945202022⋅5⋅101
Squares mod primes
n12345678910111213141516171819202122232425
n2149162536496481100121144169196225256289324361400441484529576625
mod 31101101101101101101101101
mod 51441014410144101441014410
mod 71422410142241014224101422
mod 111495335941014953359410149
mod 13149312101012394101493121010123941
mod 171491682151313152816941014916821513
mod 19149166171175571117616941014916617
mod 23149162133181286681218313216941014
mod 2914916257206231352824222224285132362072516
mod 31149162551821972820141088101420287192185
mod 3714916253612277261033211133430282830343112133
mod 411491625368234018392153220102373331313337210
mod 431491625366213814351540241041312317131111131723
mod 47149162536217346273288372174232241814121214
q
357111317192329313741434753596167717379838997
p3 NRNRNRNNRRNRNNNRRNRRNNR
5N NRNNRNRRNRNNNRRNRNRNRN
7NN RNNNRRNRNRNRNNRRNRNNN
11RRN NNNRNRRNNRRRNRRNNNRR
13RNNN RNRRNNNRNRNRNNNRNNN
17NNNNR RNNNNNRRRRNRNNNRRN
19NRRRNR RNNNNRRNNRNNRNRNN
23RNNNRNN RRNRNRNRNNRRNNNN
29NRRNRNNR NNNNNRRNRRNNRNN
31NRRNNNRNN NRNRNRNRRNNNNR
37RNRRNNNNNN RNRRNNRRRNRNN
41NRNNNNNRNRR RNNRRNNRNRNN
43NNNRRRNRNRNR RRRNRNNRRNR
47RNRNNRNNNNRNN RRRNRNRRRR
53NNRRRRNNRNRNRR RNNNNNNRR
59RRRNNRRNRNNRNNR NNRNRNNN
61RRNNRNRNNNNRNRNN NNRNRNR
67NNNNNRRRRNRNNRNRN RRNRRN
71RRNNNNRNRNRNRNNNNN RRRRN
73RNNNNNRRNNRRNNNNRRR RNRR
79NRNRRNRRNRNNNNNNNRNR RRR
83RNRRNRNRRRRRNNNRRNNNN NN
89NRNRNRNNNNNNNRRNNRRRRN R
97RNNRNNNNNRNNRRRNRNNRRNR 
q
837971675947433123191173513172937415361738997
p83 NNNRNNRRNRRRNNRRRRNRNNN
79R NRNNNRRRRNNRRNNNNNNRRR
71RR NNNRNNRNNRRNNRRNNNRRN
67RNR RRNNRRNNNNNRRRNNNRRN
59NRRN NNNNRNRRRNRRNRRNNNN
47RRRNR NNNNNRRNNRNRNRRNRR
43RRNRRR RRNRNNNRRNNRRNNNR
31NNRRRRN NRNRNRNNNNRNNNNR
23NNRNRRNR NNNRNRNRNRNNRNN
19RNNNNRRNR RRNRNRNNNNRRNN
11NNRRRRNRRN NRRNNNRNRNNRR
7NRRRNNRNRNR NNNNRRNRNNNN
3NRNRNNRRNRNR NRNNRNNRRNR
5NRRNRNNRNRRNN NNRNRNRNRN
13NRNNNNRNRNNNRN RRNNRRNNN
17RNNRRRRNNRNNNNR NNNRNNRN
29RNRRRNNNRNNRNRRN NNRNNNN
37RNRRNRNNNNRRRNNNN RRNRNN
41RNNNRNRRRNNNNRNNNR NRRNN
53NNNNRRRNNNRRNNRRRRN NNRR
61RNNNNRNNNRNNRRRNNNRN RNR
73NRRRNNNNRRNNRNNNNRRNR RR
89NRRRNRNNNNRNNRNRNNNRNR R
97NRNNNRRRNNRNRNNNNNNRRRR 

Supplements to quadratic reciprocity

The supplements to quadratic reciprocity help solve special cases of the main theorem more easily. They give simple rules for when certain numbers can be perfect squares modulo a prime number.

One important supplement deals with -1. It shows that -1 is a perfect square modulo a prime p if and only if p is congruent to 1 modulo 4. Another supplement deals with 2, showing that 2 is a perfect square modulo a prime p if and only if p is congruent to ±1 modulo 8. These supplements make it easier to work with quadratic equations in modular arithmetic without needing the full theorem.

Statement of the theorem

Quadratic reciprocity is an important idea in math. It helps us understand when certain equations have solutions. It talks about solving equations like (x^2 \equiv a \pmod{p}). In these equations, we want to know if there is a whole number (x) that makes the equation true when we only look at the remainder after dividing by a prime number (p).

The theorem gives us special rules. For example, if we have two special numbers (p) and (q), and both follow certain patterns when divided by 4, we can swap them in the equation. This still helps us know if a solution exists. This helps mathematicians understand these kinds of problems better, even if it doesn’t tell us how to find the actual solutions.

Proof

Main article: Proofs of quadratic reciprocity

Quadratic reciprocity is an important idea in math. It helps us understand patterns with numbers. One part of this is finding out if certain equations have answers. For example, we might want to know if there is a number that, when multiplied by itself, gives another number. We look at these numbers only up to a certain point, called a modulus.

There are many ways to show why quadratic reciprocity works, and new ways are still being found. These help us see connections between numbers and solve tricky problems.

History and alternative statements

The law of quadratic reciprocity is an important idea in number theory. It helps us understand when certain equations have solutions with whole numbers. Many mathematicians studied this idea over time.

Part of Article 131 in the first edition (1801) of the Disquisitiones, listing the 8 cases of quadratic reciprocity

Fermat, an early mathematician, studied prime numbers using simple equations. Euler and Lagrange also worked on similar ideas but did not fully state the law of quadratic reciprocity. Later, Legendre created a special symbol to make these ideas easier to use. Gauss, a famous mathematician, gave us the modern form of the law.

Today, we use these ideas to solve problems about whether a number can be written as a square of another number, but only in special cases involving prime numbers. This law is still important in more advanced areas of mathematics.

Main article: Legendre symbol

TheoremWhenit follows that
Ib a − 1 2 ≡ 1 ( mod a ) {\displaystyle b^{\frac {a-1}{2}}\equiv 1{\pmod {a}}} a b − 1 2 ≡ 1 ( mod b ) {\displaystyle a^{\frac {b-1}{2}}\equiv 1{\pmod {b}}}
IIa b − 1 2 ≡ − 1 ( mod b ) {\displaystyle a^{\frac {b-1}{2}}\equiv -1{\pmod {b}}} b a − 1 2 ≡ − 1 ( mod a ) {\displaystyle b^{\frac {a-1}{2}}\equiv -1{\pmod {a}}}
IIIa A − 1 2 ≡ 1 ( mod A ) {\displaystyle a^{\frac {A-1}{2}}\equiv 1{\pmod {A}}} A a − 1 2 ≡ 1 ( mod a ) {\displaystyle A^{\frac {a-1}{2}}\equiv 1{\pmod {a}}}
IVa A − 1 2 ≡ − 1 ( mod A ) {\displaystyle a^{\frac {A-1}{2}}\equiv -1{\pmod {A}}} A a − 1 2 ≡ − 1 ( mod a ) {\displaystyle A^{\frac {a-1}{2}}\equiv -1{\pmod {a}}}
Va b − 1 2 ≡ 1 ( mod b ) {\displaystyle a^{\frac {b-1}{2}}\equiv 1{\pmod {b}}} b a − 1 2 ≡ 1 ( mod a ) {\displaystyle b^{\frac {a-1}{2}}\equiv 1{\pmod {a}}}
VIb a − 1 2 ≡ − 1 ( mod a ) {\displaystyle b^{\frac {a-1}{2}}\equiv -1{\pmod {a}}} a b − 1 2 ≡ − 1 ( mod b ) {\displaystyle a^{\frac {b-1}{2}}\equiv -1{\pmod {b}}}
VIIb B − 1 2 ≡ 1 ( mod B ) {\displaystyle b^{\frac {B-1}{2}}\equiv 1{\pmod {B}}} B b − 1 2 ≡ − 1 ( mod b ) {\displaystyle B^{\frac {b-1}{2}}\equiv -1{\pmod {b}}}
VIIIb B − 1 2 ≡ − 1 ( mod B ) {\displaystyle b^{\frac {B-1}{2}}\equiv -1{\pmod {B}}} B b − 1 2 ≡ 1 ( mod b ) {\displaystyle B^{\frac {b-1}{2}}\equiv 1{\pmod {b}}}
CaseIfThen
1)±a R a′±a′ R a
2)±a N a′±a′ N a
3)+a R b
−a N b		±b R a
4)+a N b
−a R b		±b N a
5)±b R a+a R b
−a N b
6)±b N a+a N b
−a R b
7)+b R b′
−b N b′		−b′ N b
+b′ R b
8)−b N b′
+b R b′		+b′ R b
−b′ N b
CaseIfThen
9)±a R A±A R a
10)±b R A+A R b
−A N b
11)+a R B±B R a
12)−a R B±B N a
13)+b R B−B N b
+N R b
14)−b R B+B R b
−B N b

Connection with cyclotomic fields

Early proofs of quadratic reciprocity were hard to understand. Things changed when Gauss showed that certain number patterns, called quadratic fields, are part of bigger patterns known as cyclotomic fields. This helped explain quadratic reciprocity better.

Later, a mathematician named Robert Langlands suggested a big idea called the Langlands program. It aims to extend our understanding even further. Langlands himself said he didn’t find quadratic reciprocity interesting at first, but later saw its importance.

Other rings

There are quadratic reciprocity laws that work in structures other than the regular whole numbers. These help us understand patterns in more complex mathematical systems.

Gaussian integers

Carl Friedrich Gauss discovered a version of quadratic reciprocity for Gaussian integers, which are numbers of the form a + b_i, where i is the square root of -1. This work was part of his studies on quartic reciprocity. Later mathematicians built on his ideas.

Eisenstein integers

Eisenstein integers are another special set of numbers, involving a special kind of cube root of unity. Similar reciprocity laws apply here.

Imaginary quadratic fields

More general versions of quadratic reciprocity exist for imaginary quadratic fields — these are number systems built around solutions to equations like _x² + y² = 1. These laws help us understand properties of these more complex number systems.

Polynomials over a finite field

Quadratic reciprocity also applies to polynomials over finite fields — sets with a limited number of elements. This helps us study patterns in polynomial equations within these systems.

Higher powers

Further information: Cubic reciprocity, Quartic reciprocity, Octic reciprocity, and Eisenstein reciprocity

In the 1800s, mathematicians tried to expand the ideas of quadratic reciprocity to work with numbers raised to higher powers. This led them to study more complex number systems.

One big unsolved problem in math, asked in 1900, was to find a general rule for these higher powers. Later, a mathematician named Emil Artin found a broad theorem called Artin reciprocity that includes all earlier rules.

This article is a child-friendly adaptation of the Wikipedia article on Quadratic reciprocity, available under CC BY-SA 4.0.

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