Quadratic reciprocity

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

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

${\displaystyle \left({\frac {q}{p}}\right)={\begin{cases}1&{\text{if }}n^{2}\equiv q{\bmod {p}}{\text{ for some integer }}n\\-1&{\text{otherwise}}\end{cases}}}$

Then:

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

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

This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form ${\displaystyle x^{2}\equiv a{\bmod {p}}}$ for an odd prime ${\displaystyle p}$; that is, to determine the "perfect squares" modulo ${\displaystyle p}$. However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case ${\displaystyle p\equiv 3{\bmod {4}}}$ using Euler's criterion one can give an explicit formula for the "square roots" modulo ${\displaystyle p}$ of a quadratic residue ${\displaystyle a}$, namely,

${\displaystyle \pm a^{\frac {p+1}{4}}}$

indeed,

${\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.}$

This formula only works if it is known in advance that ${\displaystyle a}$ is a quadratic residue, which can be checked using the law of quadratic reciprocity.

The quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss,[1] who referred to it as the "fundamental theorem" in his Disquisitiones Arithmeticae and his papers, writing

The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151)

Privately, Gauss referred to it as the "golden theorem".[2] He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs.[3] The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2).

Generalizing the reciprocity law to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program.

Motivating examples

Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, we give examples which lead to the general case.

Factoring n² − 5

Consider the polynomial ${\displaystyle f(n)=n^{2}-5}$ and its values for ${\displaystyle n\in \mathbb {N} .}$ The prime factorizations of these values are given as follows:

n	${\displaystyle f(n)}$			n	${\displaystyle f(n)}$			n	${\displaystyle f(n)}$
1	−4	−2^2		16	251	251		31	956	2^2⋅239
2	−1	−1		17	284	2^2⋅71		32	1019	1019
3	4	2^2		18	319	11⋅29		33	1084	2^2⋅271
4	11	11		19	356	2^2⋅89		34	1151	1151
5	20	2^2⋅5		20	395	5⋅79		35	1220	2^2⋅5⋅61
6	31	31		21	436	2^2⋅109		36	1291	1291
7	44	2^2⋅11		22	479	479		37	1364	2^2⋅11⋅31
8	59	59		23	524	2^2⋅131		38	1439	1439
9	76	2^2⋅19		24	571	571		39	1516	2^2⋅379
10	95	5⋅19		25	620	2^2⋅5⋅31		40	1595	5⋅11⋅29
11	116	2^2⋅29		26	671	11⋅61		41	1676	2^2⋅419
12	139	139		27	724	2^2⋅181		42	1759	1759
13	164	2^2⋅41		28	779	19⋅41		43	1844	2^2⋅461
14	191	191		29	836	2^2⋅11⋅19		44	1931	1931
15	220	2^2⋅5⋅11		30	895	5⋅179		45	2020	2^2⋅5⋅101

The prime factors ${\displaystyle p}$ dividing ${\displaystyle f(n)}$ are ${\displaystyle p=2,5}$, and every prime whose final digit is ${\displaystyle 1}$ or ${\displaystyle 9}$; no primes ending in ${\displaystyle 3}$ or ${\displaystyle 7}$ ever appear. Now, ${\displaystyle p}$ is a prime factor of some ${\displaystyle n^{2}-5}$ whenever ${\displaystyle n^{2}-5\equiv 0{\bmod {p}}}$, i.e. whenever ${\displaystyle n^{2}\equiv 5{\bmod {p}},}$ i.e. whenever 5 is a quadratic residue modulo ${\displaystyle p}$. This happens for ${\displaystyle p=2,5}$ and those primes with ${\displaystyle p\equiv 1,4{\bmod {5}},}$ and the latter numbers ${\displaystyle 1=(\pm 1)^{2}}$ and ${\displaystyle 4=(\pm 2)^{2}}$ are precisely the quadratic residues modulo ${\displaystyle 5}$. Therefore, except for ${\displaystyle p=2,5}$, we have that ${\displaystyle 5}$ is a quadratic residue modulo ${\displaystyle p}$ iff ${\displaystyle p}$ is a quadratic residue modulo ${\displaystyle 5}$.

The law of quadratic reciprocity gives a similar characterization of prime divisors of ${\displaystyle f(n)=n^{2}-q}$ for any prime q, which leads to a characterization for any integer ${\displaystyle q}$.

Patterns among quadratic residues

Let p be an odd prime. A number modulo p is a quadratic residue whenever it is congruent to a square (mod p); otherwise it is a quadratic non-residue. ("Quadratic" can be dropped if it is clear from the context.) Here we exclude zero as a special case. Then as a consequence of the fact that the multiplicative group of a finite field of order p is cyclic of order p-1, the following statements hold:

• There are an equal number of quadratic residues and non-residues; and
• The product of two quadratic residues is a residue, the product of a residue and a non-residue is a non-residue, and the product of two non-residues is a residue.

For the avoidance of doubt, these statements do not hold if the modulus is not prime. For example, there are only 3 quadratic residues (1, 4 and 9) in the multiplicative group modulo 15. Moreover although 7 and 8 are quadratic non-residues, their product 7x8 = 11 is also a quadratic non-residue, in contrast to the prime case.

Quadratic residues are entries in the following table:

Squares mod primes

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
n^2	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256	289	324	361	400	441	484	529	576	625
mod 3	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1	1	0	1
mod 5	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0	1	4	4	1	0
mod 7	1	4	2	2	4	1	0	1	4	2	2	4	1	0	1	4	2	2	4	1	0	1	4	2	2
mod 11	1	4	9	5	3	3	5	9	4	1	0	1	4	9	5	3	3	5	9	4	1	0	1	4	9
mod 13	1	4	9	3	12	10	10	12	3	9	4	1	0	1	4	9	3	12	10	10	12	3	9	4	1
mod 17	1	4	9	16	8	2	15	13	13	15	2	8	16	9	4	1	0	1	4	9	16	8	2	15	13
mod 19	1	4	9	16	6	17	11	7	5	5	7	11	17	6	16	9	4	1	0	1	4	9	16	6	17
mod 23	1	4	9	16	2	13	3	18	12	8	6	6	8	12	18	3	13	2	16	9	4	1	0	1	4
mod 29	1	4	9	16	25	7	20	6	23	13	5	28	24	22	22	24	28	5	13	23	6	20	7	25	16
mod 31	1	4	9	16	25	5	18	2	19	7	28	20	14	10	8	8	10	14	20	28	7	19	2	18	5
mod 37	1	4	9	16	25	36	12	27	7	26	10	33	21	11	3	34	30	28	28	30	34	3	11	21	33
mod 41	1	4	9	16	25	36	8	23	40	18	39	21	5	32	20	10	2	37	33	31	31	33	37	2	10
mod 43	1	4	9	16	25	36	6	21	38	14	35	15	40	24	10	41	31	23	17	13	11	11	13	17	23
mod 47	1	4	9	16	25	36	2	17	34	6	27	3	28	8	37	21	7	42	32	24	18	14	12	12	14

This table is complete for odd primes less than 50. To check whether a number m is a quadratic residue mod one of these primes p, find a ≡ m (mod p) and 0 ≤ a < p. If a is in row p, then m is a residue (mod p); if a is not in row p of the table, then m is a nonresidue (mod p).

The quadratic reciprocity law is the statement that certain patterns found in the table are true in general.

q = ±1 and the first supplement

Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. The former set of primes are all congruent to 1 modulo 4, and the latter are congruent to 3 modulo 4.

First Supplement to Quadratic Reciprocity. The congruence ${\displaystyle x^{2}\equiv -1{\bmod {p}}}$ is solvable if and only if ${\displaystyle p}$ is congruent to 1 modulo 4.

q = ±2 and the second supplement

Examining the table, we find 2 in rows 7, 17, 23, 31, 41, and 47, but not in rows 3, 5, 11, 13, 19, 29, 37, or 43. The former primes are all ≡ ±1 (mod 8), and the latter are all ≡ ±3 (mod 8). This leads to

Second Supplement to Quadratic Reciprocity. The congruence ${\displaystyle x^{2}\equiv 2{\bmod {p}}}$ is solvable if and only if ${\displaystyle p}$ is congruent to ±1 modulo 8.

−2 is in rows 3, 11, 17, 19, 41, 43, but not in rows 5, 7, 13, 23, 29, 31, 37, or 47. The former are ≡ 1 or ≡ 3 (mod 8), and the latter are ≡ 5, 7 (mod 8).

q = ±3

3 is in rows 11, 13, 23, 37, and 47, but not in rows 5, 7, 17, 19, 29, 31, 41, or 43. The former are ≡ ±1 (mod 12) and the latter are all ≡ ±5 (mod 12).

−3 is in rows 7, 13, 19, 31, 37, and 43 but not in rows 5, 11, 17, 23, 29, 41, or 47. The former are ≡ 1 (mod 3) and the latter ≡ 2 (mod 3).

Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue modulo 3.

q = ±5

5 is in rows 11, 19, 29, 31, and 41 but not in rows 3, 7, 13, 17, 23, 37, 43, or 47. The former are ≡ ±1 (mod 5) and the latter are ≡ ±2 (mod 5).

Since the only residues (mod 5) are ±1, we see that 5 is a quadratic residue modulo every prime which is a residue modulo 5.

−5 is in rows 3, 7, 23, 29, 41, 43, and 47 but not in rows 11, 13, 17, 19, 31, or 37. The former are ≡ 1, 3, 7, 9 (mod 20) and the latter are ≡ 11, 13, 17, 19 (mod 20).

Higher q

The observations about −3 and 5 continue to hold: −7 is a residue modulo p if and only if p is a residue modulo 7, −11 is a residue modulo p if and only if p is a residue modulo 11, 13 is a residue (mod p) if and only if p is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which depend upon congruences modulo 12 and 20 respectively, are simply the ones for −3 and 5 working with the first supplement.

Example. For −5 to be a residue (mod p), either both 5 and −1 have to be residues (mod p) or they both have to be non-residues: i.e., p ≡ ±1 (mod 5) and p ≡ 1 (mod 4) or p ≡ ±2 (mod 5) and p ≡ 3 (mod 4). Using the Chinese remainder theorem these are equivalent to p ≡ 1, 9 (mod 20) or p ≡ 3, 7 (mod 20).

The generalization of the rules for −3 and 5 is Gauss's statement of quadratic reciprocity.

Legendre's version

Another way to organize the data is to see which primes are residues mod which other primes, as illustrated in the following table. The entry in row p column q is R if q is a quadratic residue (mod p); if it is a nonresidue the entry is N.

If the row, or the column, or both, are ≡ 1 (mod 4) the entry is blue or green; if both row and column are ≡ 3 (mod 4), it is yellow or orange.

The blue and green entries are symmetric around the diagonal: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is R (resp N).

The yellow and orange ones, on the other hand, are antisymmetric: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is N (resp R).

The reciprocity law states that these patterns hold for all p and q.

Legend

R	q is a residue (mod p)	q ≡ 1 (mod 4) or p ≡ 1 (mod 4) (or both)
N	q is a nonresidue (mod p)
R	q is a residue (mod p)	both q ≡ 3 (mod 4) and p ≡ 3 (mod 4)
N	q is a nonresidue (mod p)

		q
		3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
p	3		N	R	N	R	N	R	N	N	R	R	N	R	N	N	N	R	R	N	R	R	N	N	R
	5	N		N	R	N	N	R	N	R	R	N	R	N	N	N	R	R	N	R	N	R	N	R	N
	7	N	N		R	N	N	N	R	R	N	R	N	R	N	R	N	N	R	R	N	R	N	N	N
	11	R	R	N		N	N	N	R	N	R	R	N	N	R	R	R	N	R	R	N	N	N	R	R
	13	R	N	N	N		R	N	R	R	N	N	N	R	N	R	N	R	N	N	N	R	N	N	N
	17	N	N	N	N	R		R	N	N	N	N	N	R	R	R	R	N	R	N	N	N	R	R	N
	19	N	R	R	R	N	R		R	N	N	N	N	R	R	N	N	R	N	N	R	N	R	N	N
	23	R	N	N	N	R	N	N		R	R	N	R	N	R	N	R	N	N	R	R	N	N	N	N
	29	N	R	R	N	R	N	N	R		N	N	N	N	N	R	R	N	R	R	N	N	R	N	N
	31	N	R	R	N	N	N	R	N	N		N	R	N	R	N	R	N	R	R	N	N	N	N	R
	37	R	N	R	R	N	N	N	N	N	N		R	N	R	R	N	N	R	R	R	N	R	N	N
	41	N	R	N	N	N	N	N	R	N	R	R		R	N	N	R	R	N	N	R	N	R	N	N
	43	N	N	N	R	R	R	N	R	N	R	N	R		R	R	R	N	R	N	N	R	R	N	R
	47	R	N	R	N	N	R	N	N	N	N	R	N	N		R	R	R	N	R	N	R	R	R	R
	53	N	N	R	R	R	R	N	N	R	N	R	N	R	R		R	N	N	N	N	N	N	R	R
	59	R	R	R	N	N	R	R	N	R	N	N	R	N	N	R		N	N	R	N	R	N	N	N
	61	R	R	N	N	R	N	R	N	N	N	N	R	N	R	N	N		N	N	R	N	R	N	R
	67	N	N	N	N	N	R	R	R	R	N	R	N	N	R	N	R	N		R	R	N	R	R	N
	71	R	R	N	N	N	N	R	N	R	N	R	N	R	N	N	N	N	N		R	R	R	R	N
	73	R	N	N	N	N	N	R	R	N	N	R	R	N	N	N	N	R	R	R		R	N	R	R
	79	N	R	N	R	R	N	R	R	N	R	N	N	N	N	N	N	N	R	N	R		R	R	R
	83	R	N	R	R	N	R	N	R	R	R	R	R	N	N	N	R	R	N	N	N	N		N	N
	89	N	R	N	R	N	R	N	N	N	N	N	N	N	R	R	N	N	R	R	R	R	N		R
	97	R	N	N	R	N	N	N	N	N	R	N	N	R	R	R	N	R	N	N	R	R	N	R

Statement of the theorem

Quadratic Reciprocity (Gauss's statement). If ${\displaystyle q\equiv 1{\bmod {4}}}$ then the congruence ${\displaystyle x^{2}\equiv p{\bmod {q}}}$ is solvable if and only if ${\displaystyle x^{2}\equiv q{\bmod {p}}}$ is solvable. If ${\displaystyle q\equiv 3{\bmod {4}}}$ then the congruence ${\displaystyle x^{2}\equiv p{\bmod {q}}}$ is solvable if and only if ${\displaystyle x^{2}\equiv -q{\bmod {p}}}$ is solvable.

Quadratic Reciprocity (combined statement). Define ${\displaystyle q^{*}=(-1)^{\frac {q-1}{2}}q}$. Then the congruence ${\displaystyle x^{2}\equiv p{\bmod {q}}}$ is solvable if and only if ${\displaystyle x^{2}\equiv q^{*}{\bmod {p}}}$ is solvable.

Quadratic Reciprocity (Legendre's statement). If p or q are congruent to 1 modulo 4, then: ${\displaystyle x^{2}\equiv q{\bmod {p}}}$ is solvable if and only if ${\displaystyle x^{2}\equiv p{\bmod {q}}}$ is solvable. If p and q are congruent to 3 modulo 4, then: ${\displaystyle x^{2}\equiv q{\bmod {p}}}$ is solvable if and only if ${\displaystyle x^{2}\equiv p{\bmod {q}}}$ is not solvable.

The last is immediately equivalent to the modern form stated in the introduction above. It is a simple exercise to prove that Legendre's and Gauss's statements are equivalent – it requires no more than the first supplement and the facts about multiplying residues and nonresidues.

Proof

The following proof, from The American Mathematical Monthly,[4] is apparently the shortest one known.

Let

${\displaystyle S_{n}(t)=\sum _{x_{1}+x_{2}+\dots +x_{n}=t}\left({\frac {x_{1}x_{2}\cdots x_{n}}{p}}\right),}$

where ${\displaystyle t,x_{i}\in \mathbb {Z} _{p}}$ and ${\displaystyle \left({\frac {a}{p}}\right)}$ is the Legendre symbol. For an odd ${\displaystyle n}$ and any ${\displaystyle a\in \mathbb {Z} _{p}/\{0\},}$

${\displaystyle S_{n}(t)=\left({\frac {a}{p}}\right)^{n}\sum _{{\frac {x_{1}}{a}}+\cdots +{\frac {x_{n}}{a}}={\frac {t}{a}}}\left({\frac {{\frac {x_{1}}{a}}\cdots {\frac {x_{n}}{a}}}{p}}\right)=\left({\frac {a}{p}}\right)S_{n}(t/a).}$

In particular, substituting ${\displaystyle t=0}$ and ${\displaystyle a}$ a nonresidue, we get ${\displaystyle S_{n}(0)=0}$, and setting ${\displaystyle t=a}$, we get ${\displaystyle S_{n}(a)=\left({\frac {a}{p}}\right)S_{n}(1)}$; and by similar reasoning,

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

Furthermore,

${\displaystyle S_{2}(0)=\sum _{x\in \mathbb {Z} _{p}/\{0\}}\left({\frac {-x^{2}}{p}}\right)=\left({\frac {-1}{p}}\right)(p-1),}$

and, recalling that

${\displaystyle \sum _{t\in \mathbb {Z} _{p}}\left({\frac {t}{p}}\right)=0,}$

${\displaystyle S_{2}(1)=\sum _{x\in \mathbb {Z} _{p}/\{0\}}\left({\frac {x^{2}x^{-1}(1-x)}{p}}\right)=\sum _{x\in \mathbb {Z} _{p}/\{0\},\ y=x^{-1}}\left({\frac {y-1}{p}}\right)=-\left({\frac {-1}{p}}\right).}$

Therefore, for odd ${\displaystyle n}$ we have

${\displaystyle S_{n+2}(1)=\sum _{t\in \mathbb {Z} _{p}/\{0,1\}}S_{n}(t)S_{2}(1-t)+S_{n}(1)S_{2}(0)+S_{n}(0)S_{2}(1)=\sum _{t\in \mathbb {Z} _{p}/\{0,1\}}\left({\frac {t}{p}}\right)S_{n}(1)\left(-\left({\frac {-1}{p}}\right)\right)+S_{n}(1)\left({\frac {-1}{p}}\right)(p-1)=S_{n}(1)\left({\frac {-1}{p}}\right)p.}$

Since ${\displaystyle S_{1}(1)=\left({\frac {1}{p}}\right)=1}$, by induction for odd ${\displaystyle n}$

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

Therefore, by Euler's criterion, for an odd prime ${\displaystyle q}$,

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

Now, the ${\displaystyle q}$ cyclic shifts of a given ${\displaystyle q}$-tuple ${\displaystyle (x_{1},\dots ,x_{q})}$ are distinct unless all ${\displaystyle x_{i}}$ are equal, since the period of its repeated single-position cyclic shift divides ${\displaystyle q}$, and so is ${\displaystyle q}$ or 1. When they are distinct, their total contribution to the sum defining ${\displaystyle S_{q}(1)}$ is ${\displaystyle q\left({\frac {x_{1}x_{2}\cdots x_{q}}{p}}\right)}$, which is divisible by ${\displaystyle q}$. Therefore, modulo ${\displaystyle q}$ (we take ${\displaystyle q\neq p}$),

${\displaystyle S_{q}(1)\equiv \sum _{x_{1}+x_{2}+\dots +x_{q}=1,\ x_{1}=x_{2}=\dots =x_{q}}\left({\frac {x_{1}x_{2}\cdots x_{q}}{p}}\right)=\left({\frac {q^{-1}}{p}}\right)^{q}=\left({\frac {q}{p}}\right).}$

So

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

and ${\displaystyle \left({\frac {q}{p}}\right)}$ are congruent to ${\displaystyle S_{q}(1)}$, and thus to each other, modulo ${\displaystyle q}$ – but they both are numbers of the form ${\displaystyle \pm 1}$, so they are equal, which is the law of quadratic reciprocity.

Proofs of the supplements

The value of the Legendre symbol of ${\displaystyle -1}$ (used in the proof above) follows directly from Euler's criterion:

${\displaystyle \left({\frac {-1}{p}}\right)\equiv (-1)^{\frac {p-1}{2}}{\bmod {p}}}$

by Euler's criterion, but both sides of this congruence are numbers of the form ${\displaystyle \pm 1}$, so they must be equal.

Whether ${\displaystyle 2}$ is a quadratic residue can be concluded if we know the number of solutions of the equation ${\displaystyle x^{2}+y^{2}=2}$ with ${\displaystyle x,y\in \mathbb {Z} _{p},}$ which can be solved by standard methods. Namely, all its solutions where ${\displaystyle xy\neq 0,x\neq \pm y}$ can be grouped into octuplets of the form ${\displaystyle (\pm x,\pm y),(\pm y,\pm x)}$, and what is left are four solutions of the form ${\displaystyle (\pm 1,\pm 1)}$ and possibly four additional solutions where ${\displaystyle x^{2}=2,y=0}$ and ${\displaystyle x=0,y^{2}=2}$, which exist precisely if ${\displaystyle 2}$ is a quadratic residue. That is, ${\displaystyle 2}$ is a quadratic residue precisely if the number of solutions of this equation is divisible by ${\displaystyle 8}$. And this equation can be solved in just the same way here as over the rational numbers: substitute ${\displaystyle x=a+1,y=at+1}$, where we demand that ${\displaystyle a\neq 0}$ (leaving out the two solutions ${\displaystyle (1,\pm 1)}$), then the original equation transforms into

${\displaystyle a=-{\frac {2(t+1)}{(t^{2}+1)}}.}$

Here ${\displaystyle t}$ can have any value that does not make the denominator zero - for which there are ${\displaystyle 1+\left({\frac {-1}{p}}\right)}$ possibilities (i.e. ${\displaystyle 2}$ if ${\displaystyle -1}$ is a residue, ${\displaystyle 0}$ if not) - and also does not make ${\displaystyle a}$ zero, which excludes one more option, ${\displaystyle t=-1}$. Thus there are

${\displaystyle p-\left(1+\left({\frac {-1}{p}}\right)\right)-1}$

possibilities for ${\displaystyle t}$, and so together with the two excluded solutions there are overall ${\displaystyle p-\left({\frac {-1}{p}}\right)}$ solutions of the original equation. Therefore, ${\displaystyle 2}$ is a residue modulo ${\displaystyle p}$ if and only if ${\displaystyle 8}$ divides ${\displaystyle p-(-1)^{\frac {p-1}{2}}}$. This is a reformulation of the condition stated above.

History and alternative statements

The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol.

In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers.

Fermat

Fermat proved[5] (or claimed to have proved)[6] a number of theorems about expressing a prime by a quadratic form:

${\displaystyle {\begin{aligned}p=x^{2}+y^{2}\qquad &\Longleftrightarrow \qquad p=2\quad {\text{ or }}\quad p\equiv 1{\bmod {4}}\\p=x^{2}+2y^{2}\qquad &\Longleftrightarrow \qquad p=2\quad {\text{ or }}\quad p\equiv 1,3{\bmod {8}}\\p=x^{2}+3y^{2}\qquad &\Longleftrightarrow \qquad p=3\quad {\text{ or }}\quad p\equiv 1{\bmod {3}}\\\end{aligned}}}$

He did not state the law of quadratic reciprocity, although the cases −1, ±2, and ±3 are easy deductions from these and other of his theorems.

He also claimed to have a proof that if the prime number p ends with 7, (in base 10) and the prime number q ends in 3, and p ≡ q ≡ 3 (mod 4), then

${\displaystyle pq=x^{2}+5y^{2}.}$

Euler conjectured, and Lagrange proved, that[7]

${\displaystyle {\begin{aligned}p&\equiv 1,9{\bmod {20}}\quad \Longrightarrow \quad p=x^{2}+5y^{2}\\p,q&\equiv 3,7{\bmod {20}}\quad \Longrightarrow \quad pq=x^{2}+5y^{2}\end{aligned}}}$

Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem.

Euler

Translated into modern notation, Euler stated [8] that for distinct odd primes p and q:

1. If q ≡ 1 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b such that p ≡ b² (mod q).
2. If q ≡ 3 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b which is odd and not divisible by q such that p ≡ ±b² (mod 4q).

This is equivalent to quadratic reciprocity.

He could not prove it, but he did prove the second supplement.[9]

Legendre and his symbol

Fermat proved that if p is a prime number and a is an integer,

${\displaystyle a^{p}\equiv a{\bmod {p}}.}$

Thus if p does not divide a, using the non-obvious fact (see for example Ireland and Rosen below) that the residues modulo p form a field and therefore in particular the multiplicative group is cyclic, hence there can be at most two solutions to a quadratic equation:

${\displaystyle a^{\frac {p-1}{2}}\equiv \pm 1{\bmod {p}}.}$

Legendre[10] lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity:

Theorem	When	it follows that
I	${\displaystyle b^{\frac {a-1}{2}}\equiv 1{\bmod {a}}}$	${\displaystyle a^{\frac {b-1}{2}}\equiv 1{\bmod {b}}}$
II	${\displaystyle a^{\frac {b-1}{2}}\equiv -1{\bmod {b}}}$	${\displaystyle b^{\frac {a-1}{2}}\equiv -1{\bmod {a}}}$
III	${\displaystyle a^{\frac {A-1}{2}}\equiv 1{\bmod {A}}}$	${\displaystyle A^{\frac {a-1}{2}}\equiv 1{\bmod {a}}}$
IV	${\displaystyle a^{\frac {A-1}{2}}\equiv -1{\bmod {A}}}$	${\displaystyle A^{\frac {a-1}{2}}\equiv -1{\bmod {a}}}$
V	${\displaystyle a^{\frac {b-1}{2}}\equiv 1{\bmod {b}}}$	${\displaystyle b^{\frac {a-1}{2}}\equiv 1{\bmod {a}}}$
VI	${\displaystyle b^{\frac {a-1}{2}}\equiv -1{\bmod {a}}}$	${\displaystyle a^{\frac {b-1}{2}}\equiv -1{\bmod {b}}}$
VII	${\displaystyle b^{\frac {B-1}{2}}\equiv 1{\bmod {B}}}$	${\displaystyle B^{\frac {b-1}{2}}\equiv -1{\bmod {b}}}$
VIII	${\displaystyle b^{\frac {B-1}{2}}\equiv -1{\bmod {B}}}$	${\displaystyle B^{\frac {b-1}{2}}\equiv 1{\bmod {b}}}$

He says that since expressions of the form

${\displaystyle N^{\frac {c-1}{2}}{\bmod {c}},\qquad \gcd(N,c)=1}$

will come up so often he will abbreviate them as:

${\displaystyle \left({\frac {N}{c}}\right)\equiv N^{\frac {c-1}{2}}{\bmod {c}}=\pm 1.}$

This is now known as the Legendre symbol, and an equivalent[11][12] definition is used today: for all integers a and all odd primes p

${\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}0&a\equiv 0{\bmod {p}}\\1&a\not \equiv 0{\bmod {p}}{\text{ and }}\exists x:a\equiv x^{2}{\bmod {p}}\\-1&a\not \equiv 0{\bmod {p}}{\text{ and there is no such }}x.\end{cases}}}$

Legendre's version of quadratic reciprocity

${\displaystyle \left({\frac {p}{q}}\right)={\begin{cases}\left({\tfrac {q}{p}}\right)&p\equiv 1{\bmod {4}}\quad {\text{ or }}\quad q\equiv 1{\bmod {4}}\\-\left({\tfrac {q}{p}}\right)&p\equiv 3{\bmod {4}}\quad {\text{ and }}\quad q\equiv 3{\bmod {4}}\end{cases}}}$

He notes that these can be combined:

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

A number of proofs, especially those based on Gauss's Lemma,[13] explicitly calculate this formula.

The supplementary laws using Legendre symbols

${\displaystyle {\begin{aligned}\left({\frac {-1}{p}}\right)&=(-1)^{\frac {p-1}{2}}={\begin{cases}1&p\equiv 1{\bmod {4}}\\-1&p\equiv 3{\bmod {4}}\end{cases}}\\\left({\frac {2}{p}}\right)&=(-1)^{\frac {p^{2}-1}{8}}={\begin{cases}1&p\equiv 1,7{\bmod {8}}\\-1&p\equiv 3,5{\bmod {8}}\end{cases}}\end{aligned}}}$

From these two supplements, we can obtain a third reciprocity law for the quadratic character -2 as follows:

For -2 to be a quadratic residue, either 1 or -2 are both quadratic residues, or both non-residues :${\displaystyle {\bmod {p}}}$.

So either :${\displaystyle {\frac {p-1}{2}}{\text{ or }}{\frac {p^{2}-1}{8}}}$ are both even, or they are both odd. The sum of these two expressions is

${\displaystyle {\frac {p^{2}+4p-5}{8}}}$ which is even. Therefore,

${\displaystyle {\begin{aligned}\left({\frac {-2}{p}}\right)&=(-1)^{\frac {p^{2}+4p-5}{8}}={\begin{cases}1&p\equiv 1,3{\bmod {8}}\\-1&p\equiv 5,7{\bmod {8}}\end{cases}}\end{aligned}}}$

Legendre's attempt to prove reciprocity is based on a theorem of his:

Legendre's Theorem. Let a, b and c be integers where any pair of the three are relatively prime. Moreover assume that at least one of ab, bc or ca is negative (i.e. they don't all have the same sign). If

${\displaystyle {\begin{aligned}u^{2}&\equiv -bc{\bmod {a}}\\v^{2}&\equiv -ca{\bmod {b}}\\w^{2}&\equiv -ab{\bmod {c}}\end{aligned}}}$

are solvable then the following equation has a nontrivial solution in integers:

${\displaystyle ax^{2}+by^{2}+cz^{2}=0.}$

Example. Theorem I is handled by letting a ≡ 1 and b ≡ 3 (mod 4) be primes and assuming that ${\displaystyle \left({\tfrac {b}{a}}\right)=1}$ and, contrary the theorem, that ${\displaystyle \left({\tfrac {a}{b}}\right)=-1.}$ Then ${\displaystyle x^{2}+ay^{2}-bz^{2}=0}$ has a solution, and taking congruences (mod 4) leads to a contradiction.

This technique doesn't work for Theorem VIII. Let b ≡ B ≡ 3 (mod 4), and assume

${\displaystyle \left({\frac {B}{b}}\right)=\left({\frac {b}{B}}\right)=-1.}$

Then if there is another prime p ≡ 1 (mod 4) such that

${\displaystyle \left({\frac {p}{b}}\right)=\left({\frac {p}{B}}\right)=-1,}$

the solvability of ${\displaystyle Bx^{2}+by^{2}-pz^{2}=0}$ leads to a contradiction (mod 4). But Legendre was unable to prove there has to be such a prime p; he was later able to show that all that is required is:

Legendre's Lemma. If p is a prime that is congruent to 1 modulo 4 then there exists an odd prime q such that ${\displaystyle \left({\tfrac {p}{q}}\right)=-1.}$

but he couldn't prove that either. Hilbert symbol (below) discusses how techniques based on the existence of solutions to ${\displaystyle ax^{2}+by^{2}+cz^{2}=0}$ can be made to work.

Gauss

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

Gauss first proves[14] the supplementary laws. He sets[15] the basis for induction by proving the theorem for ±3 and ±5. Noting[16] that it is easier to state for −3 and +5 than it is for +3 or −5, he states[17] the general theorem in the form:

If p is a prime of the form 4n + 1 then p, but if p is of the form 4n + 3 then −p, is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p. In the next sentence, he christens it the "fundamental theorem" (Gauss never used the word "reciprocity").

Introducing the notation a R b (resp. a N b) to mean a is a quadratic residue (resp. nonresidue) (mod b), and letting a, a′, etc. represent positive primes ≡ 1 (mod 4) and b, b′, etc. positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre:

Case	If	Then
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

In the next Article he generalizes this to what are basically the rules for the Jacobi symbol (below). Letting A, A′, etc. represent any (prime or composite) positive numbers ≡ 1 (mod 4) and B, B′, etc. positive numbers ≡ 3 (mod 4):

Case	If	Then
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

All of these cases take the form "if a prime is a residue (mod a composite), then the composite is a residue or nonresidue (mod the prime), depending on the congruences (mod 4)". He proves that these follow from cases 1) - 8).

Gauss needed, and was able to prove,[18] a lemma similar to the one Legendre needed:

Gauss's Lemma. If p is a prime congruent to 1 modulo 8 then there exists an odd prime q such that:

${\displaystyle q<2{\sqrt {p}}+1\quad {\text{and}}\quad \left({\frac {p}{q}}\right)=-1.}$

The proof of quadratic reciprocity uses complete induction.

Gauss's Version in Legendre Symbols.

${\displaystyle \left({\frac {p}{q}}\right)={\begin{cases}\left({\frac {q}{p}}\right)&q\equiv 1{\bmod {4}}\\\left({\frac {-q}{p}}\right)&q\equiv 3{\bmod {4}}\end{cases}}}$

These can be combined:

Gauss's Combined Version in Legendre Symbols. Let

${\displaystyle q^{*}=(-1)^{\frac {q-1}{2}}q.}$

In other words:

${\displaystyle |q^{*}|=|q|\quad {\text{and}}\quad q^{*}\equiv 1{\bmod {4}}.}$

Then:

${\displaystyle \left({\frac {p}{q}}\right)=\left({\frac {q^{*}}{p}}\right).}$

A number of proofs of the theorem, especially those based on Gauss sums derive this formula.[19] or the splitting of primes in algebraic number fields,[20]

Other statements

The statements in this section are equivalent to quadratic reciprocity: if, for example, Euler's version is assumed, the Legendre-Gauss version can be deduced from it, and vice versa.

Euler's Formulation of Quadratic Reciprocity.[21] If ${\displaystyle p\equiv \pm q{\bmod {4a}}}$ then ${\displaystyle \left({\tfrac {a}{p}}\right)=\left({\tfrac {a}{q}}\right).}$

This can be proven using Gauss's lemma.

Quadratic Reciprocity (Gauss; Fourth Proof).[22] Let a, b, c, ... be unequal positive odd primes, whose product is n, and let m be the number of them that are ≡ 3 (mod 4); check whether n/a is a residue of a, whether n/b is a residue of b, .... The number of nonresidues found will be even when m ≡ 0, 1 (mod 4), and it will be odd if m ≡ 2, 3 (mod 4).

Gauss's fourth proof consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes. He then gives an example: Let a = 3, b = 5, c = 7, and d = 11. Three of these, 3, 7, and 11 ≡ 3 (mod 4), so m ≡ 3 (mod 4). 5×7×11 R 3; 3×7×11 R 5; 3×5×11 R 7;  and  3×5×7 N 11, so there are an odd number of nonresidues.

Eisenstein's Formulation of Quadratic Reciprocity.[23] Assume

${\displaystyle p\neq q,\quad p'\neq q',\quad p\equiv p'{\bmod {4}},\quad q\equiv q'{\bmod {4}}.}$

Then

${\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=\left({\frac {p'}{q'}}\right)\left({\frac {q'}{p'}}\right).}$

Mordell's Formulation of Quadratic Reciprocity.[24] Let a, b and c be integers. For every prime, p, dividing abc if the congruence

${\displaystyle ax^{2}+by^{2}+cz^{2}\equiv 0{\bmod {\tfrac {4abc}{p}}}}$

has a nontrivial solution, then so does:

${\displaystyle ax^{2}+by^{2}+cz^{2}\equiv 0{\bmod {4abc}}.}$

Zeta function formulation

As mentioned in the article on Dedekind zeta functions, quadratic reciprocity is equivalent to the zeta function of a quadratic field being the product of the Riemann zeta function and a certain Dirichlet L-function

Jacobi symbol

The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular

${\displaystyle {\begin{aligned}\left({\frac {-1}{n}}\right)=(-1)^{\frac {n-1}{2}}&={\begin{cases}1&n\equiv 1{\bmod {4}}\\-1&n\equiv 3{\bmod {4}}\end{cases}}\\\left({\frac {2}{n}}\right)=(-1)^{\frac {n^{2}-1}{8}}&={\begin{cases}1&n\equiv 1,7{\bmod {8}}\\-1&n\equiv 3,5{\bmod {8}}\end{cases}}\\\left({\frac {-2}{n}}\right)=(-1)^{\frac {n^{2}+4n-5}{8}}&={\begin{cases}1&n\equiv 1,3{\bmod {8}}\\-1&n\equiv 5,7{\bmod {8}}\end{cases}}\end{aligned}}}$

and if both numbers are positive and odd (this is sometimes called "Jacobi's reciprocity law"):

${\displaystyle \left({\frac {m}{n}}\right)=(-1)^{\frac {(m-1)(n-1)}{4}}\left({\frac {n}{m}}\right).}$

However, if the Jacobi symbol is 1 but the denominator is not a prime, it does not necessarily follow that the numerator is a quadratic residue of the denominator. Gauss's cases 9) - 14) above can be expressed in terms of Jacobi symbols:

${\displaystyle \left({\frac {M}{p}}\right)=(-1)^{\frac {(p-1)(M-1)}{4}}\left({\frac {p}{M}}\right),}$

and since p is prime the left hand side is a Legendre symbol, and we know whether M is a residue modulo p or not.

The formulas listed in the preceding section are true for Jacobi symbols as long as the symbols are defined. Euler's formula may be written

${\displaystyle \left({\frac {a}{m}}\right)=\left({\frac {a}{m\pm 4an}}\right),\qquad n\in \mathbb {Z} ,m\pm 4an>0.}$

Example.

${\displaystyle \left({\frac {2}{7}}\right)=\left({\frac {2}{15}}\right)=\left({\frac {2}{23}}\right)=\left({\frac {2}{31}}\right)=\cdots =1.}$

2 is a residue modulo the primes 7, 23 and 31:

${\displaystyle 3^{2}\equiv 2{\bmod {7}},\quad 5^{2}\equiv 2{\bmod {23}},\quad 8^{2}\equiv 2{\bmod {31}}.}$

But 2 is not a quadratic residue modulo 5, so it can't be one modulo 15. This is related to the problem Legendre had: if ${\displaystyle \left({\tfrac {a}{m}}\right)=-1,}$ then a is a non-residue modulo every prime in the arithmetic progression m + 4a, m + 8a, ..., if there are any primes in this series, but that wasn't proved until decades after Legendre.[25]

Eisenstein's formula requires relative primality conditions (which are true if the numbers are prime)

Let ${\displaystyle a,b,a',b'}$ be positive odd integers such that:

${\displaystyle {\begin{aligned}\gcd &(a,b)=\gcd(a',b')=1\\&a\equiv a'{\bmod {4}}\\&b\equiv b'{\bmod {4}}\end{aligned}}}$

Then

${\displaystyle \left({\frac {a}{b}}\right)\left({\frac {b}{a}}\right)=\left({\frac {a'}{b'}}\right)\left({\frac {b'}{a'}}\right).}$

Hilbert symbol

The quadratic reciprocity law can be formulated in terms of the Hilbert symbol ${\displaystyle (a,b)_{v}}$ where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the archimedean one and the p-adic absolute values for primes p). The Hilbert symbol ${\displaystyle (a,b)_{v}}$ is 1 or −1. It is defined to be 1 if and only if the equation ${\displaystyle ax^{2}+by^{2}=z^{2}}$ has a solution in the completion of the rationals at v other than ${\displaystyle x=y=z=0}$. The Hilbert reciprocity law states that ${\displaystyle (a,b)_{v}}$, for fixed a and b and varying v, is 1 for all but finitely many v and the product of ${\displaystyle (a,b)_{v}}$ over all v is 1. (This formally resembles the residue theorem from complex analysis.)

The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol. There is no kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in quadratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore, it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extension can rightly be considered a generalization of quadratic reciprocity to all global fields.

Connection with cyclotomic fields

The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.

Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[26]

I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers[27] that I appreciated it as anything more.

Other rings

There are also quadratic reciprocity laws in rings other than the integers.

Gaussian integers

In his second monograph on quartic reciprocity[28] Gauss stated quadratic reciprocity for the ring ${\displaystyle \mathbb {Z} [i]}$ of Gaussian integers, saying that it is a corollary of the biquadratic law in ${\displaystyle \mathbb {Z} [i],}$ but did not provide a proof of either theorem. Dirichlet[29] showed that the law in ${\displaystyle \mathbb {Z} [i]}$ can be deduced from the law for ${\displaystyle \mathbb {Z} }$ without using quartic reciprocity.

For an odd Gaussian prime ${\displaystyle \pi }$ and a Gaussian integer ${\displaystyle \alpha }$ relatively prime to ${\displaystyle \pi ,}$ define the quadratic character for ${\displaystyle \mathbb {Z} [i]}$ by:

${\displaystyle \left[{\frac {\alpha }{\pi }}\right]_{2}\equiv \alpha ^{\frac {\mathrm {N} \pi -1}{2}}{\bmod {\pi }}={\begin{cases}1&\exists \eta \in \mathbb {Z} [i]:\alpha \equiv \eta ^{2}{\bmod {\pi }}\\-1&{\text{otherwise}}\end{cases}}}$

Let ${\displaystyle \lambda =a+bi,\mu =c+di}$ be distinct Gaussian primes where a and c are odd and b and d are even. Then[30]

${\displaystyle \left[{\frac {\lambda }{\mu }}\right]_{2}=\left[{\frac {\mu }{\lambda }}\right]_{2},\qquad \left[{\frac {i}{\lambda }}\right]_{2}=(-1)^{\frac {b}{2}},\qquad \left[{\frac {1+i}{\lambda }}\right]_{2}=\left({\frac {2}{a+b}}\right).}$

Eisenstein integers

Consider the following third root of unity:

${\displaystyle \omega ={\frac {-1+{\sqrt {-3}}}{2}}=e^{\frac {2\pi \imath }{3}}.}$

The ring of Eisenstein integers is ${\displaystyle \mathbb {Z} [\omega ].}$[31] For an Eisenstein prime ${\displaystyle \pi ,\mathrm {N} \pi \neq 3,}$ and an Eisenstein integer ${\displaystyle \alpha }$ with ${\displaystyle \gcd(\alpha ,\pi )=1,}$ define the quadratic character for ${\displaystyle \mathbb {Z} [\omega ]}$ by the formula

${\displaystyle \left[{\frac {\alpha }{\pi }}\right]_{2}\equiv \alpha ^{\frac {\mathrm {N} \pi -1}{2}}{\bmod {\pi }}={\begin{cases}1&\exists \eta \in \mathbb {Z} [\omega ]:\alpha \equiv \eta ^{2}{\bmod {\pi }}\\-1&{\text{otherwise}}\end{cases}}}$

Let λ = a + bω and μ = c + dω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved[32]

${\displaystyle \left[{\frac {\lambda }{\mu }}\right]_{2}\left[{\frac {\mu }{\lambda }}\right]_{2}=(-1)^{{\frac {\mathrm {N} \lambda -1}{2}}{\frac {\mathrm {N} \mu -1}{2}}},\qquad \left[{\frac {1-\omega }{\lambda }}\right]_{2}=\left({\frac {a}{3}}\right),\qquad \left[{\frac {2}{\lambda }}\right]_{2}=\left({\frac {2}{\mathrm {N} \lambda }}\right).}$

Imaginary quadratic fields

The above laws are special cases of more general laws that hold for the ring of integers in any imaginary quadratic number field. Let k be an imaginary quadratic number field with ring of integers ${\displaystyle {\mathcal {O}}_{k}.}$ For a prime ideal ${\displaystyle {\mathfrak {p}}\subset {\mathcal {O}}_{k}}$ with odd norm ${\displaystyle \mathrm {N} {\mathfrak {p}}}$ and ${\displaystyle \alpha \in {\mathcal {O}}_{k},}$ define the quadratic character for ${\displaystyle {\mathcal {O}}_{k}}$ as

${\displaystyle \left[{\frac {\alpha }{\mathfrak {p}}}\right]_{2}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{2}}{\bmod {\mathfrak {p}}}={\begin{cases}1&\alpha \not \in {\mathfrak {p}}{\text{ and }}\exists \eta \in {\mathcal {O}}_{k}{\text{ such that }}\alpha -\eta ^{2}\in {\mathfrak {p}}\\-1&\alpha \not \in {\mathfrak {p}}{\text{ and there is no such }}\eta \\0&\alpha \in {\mathfrak {p}}\end{cases}}}$

for an arbitrary ideal ${\displaystyle {\mathfrak {a}}\subset {\mathcal {O}}_{k}}$ factored into prime ideals ${\displaystyle {\mathfrak {a}}={\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{n}}$ define

${\displaystyle \left[{\frac {\alpha }{\mathfrak {a}}}\right]_{2}=\left[{\frac {\alpha }{{\mathfrak {p}}_{1}}}\right]_{2}\cdots \left[{\frac {\alpha }{{\mathfrak {p}}_{n}}}\right]_{2},}$

and for ${\displaystyle \beta \in {\mathcal {O}}_{k}}$ define

${\displaystyle \left[{\frac {\alpha }{\beta }}\right]_{2}=\left[{\frac {\alpha }{\beta {\mathcal {O}}_{k}}}\right]_{2}.}$

Let ${\displaystyle {\mathcal {O}}_{k}=\mathbb {Z} \omega _{1}\oplus \mathbb {Z} \omega _{2},}$ i.e. ${\displaystyle \left\{\omega _{1},\omega _{2}\right\}}$ is an integral basis for ${\displaystyle {\mathcal {O}}_{k}.}$ For ${\displaystyle \nu \in {\mathcal {O}}_{k}}$ with odd norm ${\displaystyle \mathrm {N} \nu ,}$ define (ordinary) integers a, b, c, d by the equations,

${\displaystyle {\begin{aligned}\nu \omega _{1}&=a\omega _{1}+b\omega _{2}\\\nu \omega _{2}&=c\omega _{1}+d\omega _{2}\end{aligned}}}$

and a function

${\displaystyle \chi (\nu ):=\imath ^{(b^{2}-a+2)c+(a^{2}-b+2)d+ad}.}$

If m = Nμ and n = Nν are both odd, Herglotz proved[33]

${\displaystyle \left[{\frac {\mu }{\nu }}\right]_{2}\left[{\frac {\nu }{\mu }}\right]_{2}=(-1)^{{\frac {m-1}{2}}{\frac {n-1}{2}}}\chi (\mu )^{m{\frac {n-1}{2}}}\chi (\nu )^{-n{\frac {m-1}{2}}}.}$

Also, if

${\displaystyle \mu \equiv \mu '{\bmod {4}}\quad {\text{and}}\quad \nu \equiv \nu '{\bmod {4}}}$

Then[34]

${\displaystyle \left[{\frac {\mu }{\nu }}\right]_{2}\left[{\frac {\nu }{\mu }}\right]_{2}=\left[{\frac {\mu '}{\nu '}}\right]_{2}\left[{\frac {\nu '}{\mu '}}\right]_{2}.}$

Polynomials over a finite field

Let F be a finite field with q = pⁿ elements, where p is an odd prime number and n is positive, and let F[x] be the ring of polynomials in one variable with coefficients in F. If ${\displaystyle f,g\in F[x]}$ and f is irreducible, monic, and has positive degree, define the quadratic character for F[x] in the usual manner:

${\displaystyle \left({\frac {g}{f}}\right)={\begin{cases}1&\gcd(f,g)=1{\text{ and }}\exists h,k\in F[x]{\text{ such that }}g-h^{2}=kf\\-1&\gcd(f,g)=1{\text{ and }}g{\text{ is not a square}}{\bmod {f}}\\0&\gcd(f,g)\neq 1\end{cases}}}$

If ${\displaystyle f=f_{1}\cdots f_{n}}$ is a product of monic irreducibles let

${\displaystyle \left({\frac {g}{f}}\right)=\left({\frac {g}{f_{1}}}\right)\cdots \left({\frac {g}{f_{n}}}\right).}$

Dedekind proved that if ${\displaystyle f,g\in F[x]}$ are monic and have positive degrees,[35]

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

Higher powers

The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, Carl Gustav Jakob Jacobi, Gotthold Eisenstein, Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers;[36] specifically Kummer invented ideals in order to state and prove higher reciprocity laws.

The ninth in the list of 23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the "Proof of the most general reciprocity law [f]or an arbitrary number field".[37] Building upon work by Philipp Furtwängler, Teiji Takagi, Helmut Hasse and others, Emil Artin discovered Artin reciprocity in 1923, a general theorem for which all known reciprocity laws are special cases, and proved it in 1927.[38]

See also

• Dedekind zeta function
• Rational reciprocity law
• Zolotarev's lemma

Notes

1. Gauss, DA § 4, arts 107–150
2. E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See facsimile page from Felix Klein's Development of Mathematics in the 19th century
3. See F. Lemmermeyer's chronology and bibliography of proofs in the external references
4. Veklych, Bogdan (2019). "A Minimalist Proof of the Law of Quadratic Reciprocity". The American Mathematical Monthly. 126 (10): 928. doi:10.1080/00029890.2019.1655331.
5. Lemmermeyer, pp. 2–3
6. Gauss, DA, art. 182
7. Lemmermeyer, p. 3
8. Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61
9. Ireland & Rosen, pp. 69–70. His proof is based on what are now called Gauss sums.
10. This section is based on Lemmermeyer, pp. 6–8
11. The equivalence is Euler's criterion
12. The analogue of Legendre's original definition is used for higher-power residue symbols
13. E.g. Kronecker's proof (Lemmermeyer, ex. p. 31, 1.34) is to use Gauss's lemma to establish that

    ${\displaystyle \left({\frac {p}{q}}\right)=\operatorname {sgn} \prod _{i=1}^{\frac {q-1}{2}}\prod _{k=1}^{\frac {p-1}{2}}\left({\frac {k}{p}}-{\frac {i}{q}}\right)}$

    and then switch p and q.
14. Gauss, DA, arts 108–116
15. Gauss, DA, arts 117–123
16. Gauss, DA, arts 130
17. Gauss, DA, Art 131
18. Gauss, DA, arts. 125–129
19. Because the basic Gauss sum equals ${\displaystyle {\sqrt {q^{*}}}.}$
20. Because the quadratic field ${\displaystyle \mathbb {Q} ({\sqrt {q^{*}}})}$ is a subfield of the cyclotomic field ${\displaystyle \mathbb {Q} (e^{\frac {2\pi i}{q}})}$
21. Ireland & Rosen, pp 60–61.
22. Gauss, "Summierung gewisser Reihen von besonderer Art", reprinted in Untersuchumgen uber hohere Arithmetik, pp.463–495
23. Lemmermeyer, Th. 2.28, pp 63–65
24. Lemmermeyer, ex. 1.9, p. 28
25. By Peter Gustav Lejeune Dirichlet in 1837
26. "Archived copy" (PDF). Archived from the original (PDF) on January 22, 2012. Retrieved June 27, 2013.CS1 maint: archived copy as title (link)
27. Weyl, Hermann (1998). Algebraic Theory of Numbers. ISBN 0691059179.
28. Gauss, BQ § 60
29. Dirichlet's proof is in Lemmermeyer, Prop. 5.1 p.154, and Ireland & Rosen, ex. 26 p. 64
30. Lemmermeyer, Prop. 5.1, p. 154
31. See the articles on Eisenstein integer and cubic reciprocity for definitions and notations.
32. Lemmermeyer, Thm. 7.10, p. 217
33. Lemmermeyer, Thm 8.15, p.256 ff
34. Lemmermeyer Thm. 8.18, p. 260
35. Bach & Shallit, Thm. 6.7.1
36. Lemmermeyer, p. 15, and Edwards, pp.79–80 both make strong cases that the study of higher reciprocity was much more important as a motivation than Fermat's Last Theorem was
37. Lemmermeyer, p. viii
38. Lemmermeyer, p. ix ff

References

The Disquisitiones Arithmeticae has been translated (from Latin) into English and German. The German edition includes all of Gauss's papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".

• Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9
• Gauss, Carl Friedrich; Maser, Hermann (translator into German) (1965), Untersuchungen über höhere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8

The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n".

• Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6
• Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7

These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148. German translations are in pp. 511–533 and 534–586 of Untersuchungen über höhere Arithmetik.

Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy:

Franz Lemmermeyer's Reciprocity Laws: From Euler to Eisenstein has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law.

Kenneth Ireland and Michael Rosen's A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well. Exercise 13.26 (p. 202) says it all

Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one.

• Bach, Eric; Shallit, Jeffrey (1966), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02405-5
• Edwards, Harold (1977), Fermat's Last Theorem, New York: Springer, ISBN 0-387-90230-9
• Lemmermeyer, Franz (2000), Reciprocity Laws, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696
• Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (second edition), New York: Springer, ISBN 0-387-97329-X

External links

• "Quadratic reciprocity law", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Quadratic Reciprocity Theorem from MathWorld
• A play comparing two proofs of the quadratic reciprocity law
• A proof of this theorem at PlanetMath
• A different proof at MathPages
• F. Lemmermeyer's chronology and bibliography of proofs of the Quadratic Reciprocity Law (246 proofs)