Editing Quadratic reciprocity (section)

===Gauss===
[[Image:Disquisitiones-Arithmeticae-p133.jpg|200px|thumb|Part of Article 131 in the first edition (1801) of the ''[[Disquisitiones Arithmeticae|Disquisitiones]]'', listing the 8 cases of quadratic reciprocity]]

Gauss first proves<ref>Gauss, DA, arts 108&ndash;116</ref> the supplementary laws. He sets<ref>Gauss, DA, arts 117&ndash;123</ref> the basis for induction by proving the theorem for ±3 and ±5. Noting<ref>Gauss, DA, arts 130</ref> that it is easier to state for −3 and +5 than it is for +3 or −5, he states<ref>Gauss, DA, Art 131</ref> the general theorem in the form:

:If ''p'' is a prime of the form 4''n''&nbsp;+&nbsp;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''&prime;, etc. represent positive primes ≡ 1 (mod 4) and ''b'', ''b''&prime;, etc. positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre:

{| class="wikitable"
|+ 
! width="50"|Case 
! width="80"|If 
! width="80"|Then 
|-
! 1) 
| ±''a'' R ''a''&prime; 
| ±''a''&prime; R ''a''
|-
! 2) 
| ±''a'' N ''a''&prime; 
| ±''a''&prime; N ''a''
|-
! 3) 
| +''a'' R ''b''<br>−''a'' N ''b'' 
| ±''b'' R ''a''
|-
! 4) 
| +''a'' N ''b''<br>−''a'' R ''b'' 
| ±''b'' N ''a''
|-
! 5) 
| ±''b'' R ''a'' 
| +''a'' R ''b''<br>−''a'' N ''b''
|-
! 6) 
| ±''b'' N ''a'' 
| +''a'' N ''b''<br>−''a'' R ''b''
|-
! 7) 
| +''b'' R ''b''&prime;<br>−''b'' N ''b''&prime; 
| −''b''&prime; N ''b''<br>+''b''&prime; R ''b''
|-
! 8) 
| −''b'' N ''b''&prime;<br>+''b'' R ''b''&prime; 
| +''b''&prime; R ''b''<br>−''b''&prime; N ''b''
|}

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

{| class="wikitable"
|+ 
! width="50"|Case 
! width="80"|If 
! width="80"|Then 
|-
! 9) 
| ±''a'' R ''A'' 
| ±''A'' R ''a''
|-
! 10) 
| ±''b'' R ''A'' 
| +''A'' R ''b''<br>−''A'' N ''b''
|-
! 11) 
| +''a'' R ''B'' 
| ±''B'' R ''a''
|-
! 12) 
| −''a'' R ''B'' 
| ±''B'' N ''a''
|-
! 13) 
| +''b'' R ''B'' 
| −''B'' N ''b''<br>+''N'' R ''b''
|-
! 14) 
| −''b'' R ''B'' 
| +''B'' R ''b''<br>−''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,<ref>Gauss, DA, arts. 125&ndash;129</ref> 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:
::<math>q <2\sqrt p+1 \quad \text{and} \quad \left(\frac{p}{q}\right) = -1.</math>

The proof of quadratic reciprocity uses [[Mathematical induction#Complete induction|complete induction]].

:'''Gauss's Version in Legendre Symbols.''' 
::<math>\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}</math>

These can be combined:

:'''Gauss's Combined Version in Legendre Symbols.''' Let
::<math>q^* = (-1)^{\frac{q-1}{2}}q.</math>
:In other words:
::<math>|q^*|=|q| \quad \text{and} \quad q^*\equiv 1 \bmod{4}.</math>
:Then:
::<math> \left(\frac{p}{q}\right) = \left(\frac{q^*}{p}\right).</math>

A number of proofs of the theorem, especially those based on [[Gauss sum]]s<ref>Because the basic Gauss sum equals <math>\sqrt{q^*}.</math></ref> or the splitting of primes in [[algebraic number field]]s,<ref>Because the quadratic field <math>\Q(\sqrt{q^*})</math> is a subfield of the cyclotomic field <math>\Q(e^{\frac{2\pi i}{q}})</math></ref><ref>See [[#Connection with cyclotomic fields|Connection with cyclotomic fields]] below.</ref> derive this formula.