Gauss's lemma (number theory)


Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.
It made its first appearance in Carl Friedrich Gauss's third proof of quadratic reciprocity and he proved it again in his fifth proof.

Statement of the lemma

For any odd prime let be an integer that is coprime to.
Consider the integers
and their least positive residues modulo.
Let be the number of these residues that are greater than. Then
where is the Legendre symbol.

Example

Taking = 11 and = 7, the relevant sequence of integers is
After reduction modulo 11, this sequence becomes
Three of these integers are larger than 11/2, so = 3. Correspondingly Gauss's lemma predicts that
This is indeed correct, because 7 is not a quadratic residue modulo 11.
The above sequence of residues
may also be written
In this form, the integers larger than 11/2 appear as negative numbers. It is also apparent that the absolute values of the residues are a permutation of the residues

Proof

A fairly simple proof, reminiscent of one of the simplest proofs of Fermat's little theorem, can be obtained by evaluating the product
modulo p in two different ways. On one hand it is equal to
The second evaluation takes more work. If is a nonzero residue modulo, let us define the "absolute value" of to be
Since counts those multiples which are in the latter range, and since for those multiples, is in the first range, we have
Now observe that the values are distinct for. Indeed, we have
because is coprime to.
This gives =, since and are positive least residues. But there are exactly of them, so their values are a rearrangement of the integers. Therefore,
Comparing with our first evaluation, we may cancel out the nonzero factor
and we are left with
This is the desired result, because by Euler's criterion the left hand side is just an alternative expression for the Legendre symbol.

Applications

Gauss's lemma is used in many, but by no means all, of the known proofs of quadratic reciprocity.
For example, Gotthold Eisenstein used Gauss's lemma to prove that if is an odd prime then
and used this formula to prove quadratic reciprocity. By using elliptic rather than circular functions, he proved the cubic and quartic reciprocity laws.
Leopold Kronecker used the lemma to show that
Switching and immediately gives quadratic reciprocity.
It is also used in what are probably the simplest proofs of the "second supplementary law"

Higher powers

Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of in, the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.

''n''th power residue symbol

Let k be an algebraic number field with ring of integers and let be a prime ideal. The ideal norm of is defined as the cardinality of the residue class ring. Since is prime this is a finite field, so the ideal norm is.
Assume that a primitive th root of unity and that and are coprime. Then no two distinct th roots of unity can be congruent modulo.
This can be proved by contradiction, beginning by assuming that mod,. Let such that mod, and. From the definition of roots of unity,
and dividing by gives
Letting and taking residues mod,
Since and are coprime, mod but under the assumption, one of the factors on the right must be zero. Therefore, the assumption that two distinct roots are congruent is false.
Thus the residue classes of containing the powers of are a subgroup of order of its group of units, Therefore, the order of is a multiple of, and
There is an analogue of Fermat's theorem in. If for, then
and since mod,
is well-defined and congruent to a unique th root of unity ζns.
This root of unity is called the th-power residue symbol for and is denoted by
It can be proven that
if and only if there is an such that mod.

1/''n'' systems

Let be the multiplicative group of the th roots of unity, and let be representatives of the cosets of Then is called a system mod
In other words, there are numbers in the set and this set constitutes a representative set for
The numbers, used in the original version of the lemma, are a 1/2 system.
Constructing a system is straightforward: let be a representative set for Pick any and remove the numbers congruent to from. Pick from and remove the numbers congruent to Repeat until is exhausted. Then is a system mod

The lemma for ''n''th powers

Gauss's lemma may be extended to the th power residue symbol as follows. Let be a primitive th root of unity, a prime ideal, and let be a system mod
Then for each,, there are integers, unique, and, unique, such that
and the th-power residue symbol is given by the formula
The classical lemma for the quadratic Legendre symbol is the special case,,, if, if.

Proof

The proof of the th-power lemma uses the same ideas that were used in the proof of the quadratic lemma.
The existence of the integers and, and their uniqueness and, respectively, come from the fact that is a representative set.
Assume that = =, i.e.
and
Then
Because and are coprime both sides can be divided by, giving
which, since is a system, implies and, showing that is a permutation of the set.
Then on the one hand, by the definition of the power residue symbol,
and on the other hand, since is a permutation,
so
and since for all, and are coprime, can be cancelled from both sides of the congruence,
and the theorem follows from the fact that no two distinct th roots of unity can be congruent.

Relation to the transfer in group theory

Let be the multiplicative group of nonzero residue classes in, and let be the subgroup. Consider the following coset representatives of in,
Applying the machinery of the transfer to this collection of coset representatives, we obtain the transfer homomorphism
which turns out to be the map that sends to, where and are as in the statement of the lemma. Gauss's lemma may then be viewed as a computation that explicitly identifies this homomorphism as being the quadratic residue character.