In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the law of quadratic reciprocity is a theorem about
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
that gives conditions for the solvability of
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s modulo
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. Due to its subtlety, it has many formulations, but the most standard statement is:
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
for an odd prime
; that is, to determine the "perfect squares" modulo
. However, this is a
non-constructive
In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also ...
result: it gives no help at all for finding a ''specific'' solution; for this, other methods are required. For example, in the case
using
Euler's criterion In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then
:
a^ \equiv
\begin
\;\;\,1\pmod& \text ...
one can give an explicit formula for the "square roots" modulo
of a quadratic residue
, namely,
:
indeed,
:
This formula only works if it is known in advance that
is a
quadratic residue, which can be checked using the law of quadratic reciprocity.
The quadratic reciprocity theorem was conjectured by
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Legendre and first proved by
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, who referred to it as the "fundamental theorem" in his ''
Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'' 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". He published six
proofs
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
for it, and two more were found in his posthumous papers. There are now over 240 published proofs. 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
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...
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
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, number theory, and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, culminating in
Artin reciprocity
Artin may refer to:
* Artin (name), a surname and given name, including a list of people with the name
** Artin, a variant of Harutyun
Harutyun ( hy, Õ€Õ¡Ö€Õ¸Ö‚Õ©ÕµÕ¸Ö‚Õ¶ and in Western Armenian Õ…Õ¡Ö€Õ¸Ö‚Õ©Õ«Ö‚Õ¶) also spelled Haroutioun, Harut ...
,
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
, and the
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
.
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 ''n''2 − 5
Consider the polynomial
and its values for
The prime factorizations of these values are given as follows:
The prime factors
dividing
are
, and every prime whose final digit is
or
; no primes ending in
or
ever appear. Now,
is a prime factor of some
whenever
, i.e. whenever
i.e. whenever 5 is a quadratic residue modulo
. This happens for
and those primes with
and the latter numbers
and
are precisely the quadratic residues modulo
. Therefore, except for
, we have that
is a quadratic residue modulo
iff
is a quadratic residue modulo
.
The law of quadratic reciprocity gives a similar characterization of prime divisors of
for any prime ''q'', which leads to a characterization for any integer
.
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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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:
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.
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''.
Supplements to Quadratic Reciprocity
The supplements provide solutions to specific cases of quadratic reciprocity. They are often quoted as partial results, without having to resort to the complete theorem.
''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
is solvable if and only if
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
is solvable if and only if
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
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
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.
Statement of the theorem
Quadratic Reciprocity (Gauss's statement). If
, then the congruence
is solvable if and only if
is solvable. If
and
, then the congruence
is solvable if and only if
is solvable.
Quadratic Reciprocity (combined statement). Define
. Then the congruence
is solvable if and only if
is solvable.
Quadratic Reciprocity (Legendre's statement). If ''p'' or ''q'' are congruent to 1 modulo 4, then:
is solvable if and only if
is solvable. If ''p'' and ''q'' are congruent to 3 modulo 4, then:
is solvable if and only if
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
Apparently, the shortest known proof yet was published by B. Veklych in the ''American Mathematical Monthly''.
Proofs of the supplements
The value of the Legendre symbol of
(used in the proof above) follows directly from
Euler's criterion In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then
:
a^ \equiv
\begin
\;\;\,1\pmod& \text ...
:
:
by Euler's criterion, but both sides of this congruence are numbers of the form
, so they must be equal.
Whether
is a quadratic residue can be concluded if we know the number of solutions of the equation
with
which can be solved by standard methods. Namely, all its solutions where
can be grouped into octuplets of the form
, and what is left are four solutions of the form
and possibly four additional solutions where
and
, which exist precisely if
is a quadratic residue. That is,
is a quadratic residue precisely if the number of solutions of this equation is divisible by
. And this equation can be solved in just the same way here as over the rational numbers: substitute
, where we demand that
(leaving out the two solutions
), then the original equation transforms into
:
Here
can have any value that does not make the denominator zero – for which there are
possibilities (i.e.
if
is a residue,
if not) – and also does not make
zero, which excludes one more option,
. Thus there are
:
possibilities for
, and so together with the two excluded solutions there are overall
solutions of the original equation. Therefore,
is a residue modulo
if and only if
divides
. 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 (or claimed to have proved) a number of theorems about expressing a prime by a quadratic form:
:
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
:
Euler conjectured, and Lagrange proved, that
:
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 that for distinct odd primes ''p'' and ''q'':
# 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'' ≡ ''b''
2 (mod ''q'').
# 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'' ≡ ±''b''
2 (mod 4''q'').
This is equivalent to quadratic reciprocity.
He could not prove it, but he did prove the second supplement.
Legendre and his symbol
Fermat proved that if ''p'' is a prime number and ''a'' is an integer,
:
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and therefore in particular the multiplicative group is cyclic, hence there can be at most two solutions to a quadratic equation:
:
Legendre 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:
He says that since expressions of the form
:
will come up so often he will abbreviate them as:
:
This is now known as the
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
, and an equivalent definition is used today: for all integers ''a'' and all odd primes ''p''
:
Legendre's version of quadratic reciprocity
:
He notes that these can be combined:
:
A number of proofs, especially those based on
Gauss's Lemma, explicitly calculate this formula.
The supplementary laws using Legendre symbols
:
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 :
.
So either :
are both even, or they are both odd. The sum of these two expressions is
:
which is an integer. Therefore,
:
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
::
:are solvable then the following equation has a nontrivial solution in integers:
::
Example. Theorem I is handled by letting ''a'' ≡ 1 and ''b'' ≡ 3 (mod 4) be primes and assuming that
and, contrary the theorem, that
Then
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
:
Then if there is another prime ''p'' ≡ 1 (mod 4) such that
:
the solvability of
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
but he couldn't prove that either.
Hilbert symbol (below) discusses how techniques based on the existence of solutions to
can be made to work.
Gauss
Gauss first proves the supplementary laws. He sets the basis for induction by proving the theorem for ±3 and ±5. Noting that it is easier to state for −3 and +5 than it is for +3 or −5, he states the general theorem in the form:
:If ''p'' is a prime of the form 4''n'' + 1 then ''p'', but if ''p'' is of the form 4''n'' + 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:
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):
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, 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:
::
The proof of quadratic reciprocity uses
complete induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
.
:Gauss's Version in Legendre Symbols.
::
These can be combined:
:Gauss's Combined Version in Legendre Symbols. Let
::
:In other words:
::
:Then:
::
A number of proofs of the theorem, especially those based on
Gauss sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
:G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r)
where the sum is over elements of some finite commutative ring , is a ...
s, derive this formula or the splitting of primes in
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s.
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. If
then
This can be proven using
Gauss's lemma.
:Quadratic Reciprocity (Gauss; Fourth Proof). 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. Assume
::
:Then
::
:Mordell's Formulation of Quadratic Reciprocity. Let ''a'', ''b'' and ''c'' be integers. For every prime, ''p'', dividing ''abc'' if the congruence
::
:has a nontrivial solution, then so does:
::
:Zeta function formulation
:As mentioned in the article on
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
s, 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
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
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
:
and if both numbers are positive and odd (this is sometimes called "Jacobi's reciprocity law"):
:
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:
:
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
:
Example.
:
2 is a residue modulo the primes 7, 23 and 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
then ''a'' is a non-residue modulo every prime in the arithmetic progression ''m'' + 4''a'', ''m'' + 8''a'', ..., if there ''are'' any primes in this series, but that wasn't proved until decades after Legendre.
Eisenstein's formula requires relative primality conditions (which are true if the numbers are prime)
:Let
be positive odd integers such that:
::
:Then
::
Hilbert symbol
The quadratic reciprocity law can be formulated in terms of the
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity ...
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
is 1 or −1. It is defined to be 1 if and only if the equation
has a solution in the
completion of the rationals at ''v'' other than
. The Hilbert reciprocity law states that
, for fixed ''a'' and ''b'' and varying ''v'', is 1 for all but finitely many ''v'' and the product of
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 field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s 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 sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically
:G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r)
where the sum is over elements of some finite commutative ring , is a ...
s to show that
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s are subfields of
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
s, 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
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
, which can be viewed as a vast generalization of quadratic reciprocity.
Robert Langlands
Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...
formulated the
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
, which gives a conjectural vast generalization of class field theory. He wrote:
:''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 that I appreciated it as anything more.''
Other rings
There are also quadratic reciprocity laws in
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
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s other than the integers.
Gaussian integers
In his second monograph on
quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form o ...
Gauss stated quadratic reciprocity for the ring