Eisenstein's proof of quadratic reciprocity is a simplification of Gauss's third proof. It is more geometrically intuitive and requires less technical manipulation.
The point of departure is "Eisenstein's lemma", which states that for distinct odd primes p, q,
where denotes the floor function (the largest integer less than or equal to x), and where the sum is taken over the even integers u = 2, 4, 6, ..., p−1. For example,
This result is very similar to Gauss's lemma, and can be proved in a similar fashion (proof given below).
Using this representation of (q/p), the main argument is quite elegant. The sum counts the number of lattice points with even x-coordinate in the interior of the triangle ABC in the following diagram:
Lattice point diagram
Example showing lattice points inside ABC with even x-coordinates, for p = 11 and q = 7
Because each column has an even number of points (namely q−1 points), the number of such lattice points in the region BCYX is the same modulo 2 as the number of such points in the region CZY:
The number of points with even x-coordinate inside BCYX (marked by O's) is equal modulo 2 to the number of such points in CZY (marked by X's)
Then by flipping the diagram in both axes, we see that the number of points with even x-coordinate inside CZY is the same as the number of points inside AXY having oddx-coordinates:
The number of points with even x-coordinate inside CZY is equal to the number of points with oddx-coordinate inside AXY
The conclusion is that
where μ is the total number of lattice points in the interior of AYX. Switching p and q, the same argument shows that
where ν is the number of lattice points in the interior of WYA. Since there are no lattice points on the line AY itself (because p and q are relatively prime), and since the total number of points in the rectangle WYXA is
For an even integer u in the range 1 ≤ u ≤ p−1, denote by r(u) the least positive residue of qu modulo p. (For example, for p = 11, q = 7, we allow u = 2, 4, 6, 8, 10, and the corresponding values of r(u) are 3, 6, 9, 1, 4.) The numbers (−1)r(u)r(u), again treated as least positive residues modulo p, are all even (in our running example, they are 8, 6, 2, 10, 4.) Furthermore, they are all distinct, because if (−1)r(u)r(u) ≡ (−1)r(t)r(t) (mod p), then we may divide out by q to obtain u ≡ ±t (mod p). This forces u ≡ t (mod p), because both u and t are even, whereas p is odd. Since there exactly (p−1)/2 of them and they are distinct, they must be simply a rearrangement of the even integers 2, 4, ..., p−1. Multiplying them together, we obtain
Dividing out successively by 2, 4, ..., p−1 on both sides (which is permissible since none of them are divisible by p) and rearranging, we have
On the other hand, by the definition of r(u) and the floor function,
and so since p is odd and u is even, we see that and r(u) are congruent modulo 2. Finally this shows that
The proof of Quadratic Reciprocity using Gauss sums is one of the more common and classic proofs. These proofs work by comparing computations of single values in two different ways, one using Euler's Criterion and the other Fermat's Little Theorem. As an example of how Euler's criterion is used, we can use it to give a quick proof of the first supplemental case of determining for an odd prime p: By Euler's criterion , but since both sides of the equivalence are ±1 and p is odd, we can deduce that .
Let , a primitive 8th root of unity and set . Since and we see that . Because is an algebraic integer, if p is an odd prime it makes sense to talk about it modulo p. Using Euler's criterion, it follows that
We can then say that
But we can also compute using Fermat's Little Theorem. By Fermat's Little Theorem we find that . We can evaluate this more exactly by breaking this up into two cases
These are the only options for a prime modulo 8 and both of these cases can be computed using the exponential form . We can write this succinctly for all odd primes p as
Combining these two expressions for and multiplying through by we find that . Since both and are ±1 and 2 is invertible modulo p, we can conclude that
The idea for the general proof follows the above supplemental case: Find an algebraic integer that somehow encodes the Legendre symbols for p, then find a relationship between Legendre symbols by computing the qth power of this algebraic integer modulo q in two different ways, one using Euler's criterion the other using Fermat's Little Theorem.
where is a primitive pth root of unity. This is a Quadratic Gauss Sum. A fundamental property of these Gauss sums is that
where . To put this in context of the next proof, the individual elements of the Gauss sum are in the cyclotomic field but the above formula shows that the sum itself is a generator of the unique quadratic field contained in L. Again, since the quadratic Gauss sum is an algebraic integer, we can use modular arithmetic with it. Using this fundamental formula and Euler's criterion we find that
Using Fermat's Little Theorem, we also find that , If we let a be a multiplicative inverse of , then we can rewrite this sum as using the substitution , which doesn't affect the range of the sum. Since , we can then write
Using these two expressions for , and multiplying through by gives
Since is invertible modulo q, and the Legendre symbols are either ±1, we can then conclude that
Suppose that p is an odd prime. The action takes place inside the cyclotomic field
where ζp is a primitive pthroot of unity. The basic theory of cyclotomic fields informs us that there is a canonical isomorphism
which sends the automorphism σa satisfying
to the element
(This is because the morphism of reduction from Z to Z/qZ is injective on the set of p-th roots of unity)
Now consider the subgroup H of squares of elements of G. Since G is cyclic, H has index 2 in G, so the subfield corresponding to H under the Galois correspondence must be a quadratic extension of Q. (In fact it is the unique quadratic extension of Q contained in L.) The Gaussian period theory determines which one; it turns out to be
At this point we start to see a hint of quadratic reciprocity emerging from our framework. On one hand, the image of H in
consists precisely of the (nonzero) quadratic residues modulo p. On the other hand, H is related to an attempt to take the square root of p (or possibly of −p). In other words, if now q is an odd prime (different from p), we have so far shown that
Lemmermeyer (2000) 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.
Ireland & Rosen (1990) 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.