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.
Contents
- Statement of the lemma
- Example
- Proof
- Applications
- Higher powers
- nth power residue symbol
- 1n systems
- The lemma for nth powers
- Relation to the transfer in group theory
- References
It made its first appearance in Carl Friedrich Gauss's third proof (1808) of quadratic reciprocity and he proved it again in his fifth proof (1818).
Statement of the lemma
For any odd prime p let a be an integer that is coprime to p.
Consider the integers
and their least positive residues modulo p. (These residues are all distinct, so there are (p − 1)/2 of them.)
Let n be the number of these residues that are greater than p/2. Then
where
Example
Taking p = 11 and a = 7, the relevant sequence of integers is
7, 14, 21, 28, 35.After reduction modulo 11, this sequence becomes
7, 3, 10, 6, 2.Three of these integers are larger than 11/2 (namely 6, 7 and 10), so n = 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
7, 3, 10, 6, 2may also be written
−4, 3, −1, −5, 2.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
1, 2, 3, 4, 5.Proof
Any textbook on elementary number theory will have a proof of the lemma. 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 x is a nonzero residue modulo p, let us define the "absolute value" of x to be
Since n counts those multiples ka which are in the latter range, and since for those multiples, −ka is in the first range, we have
Now observe that the values |ra| are distinct for r = 1, 2, …, (p − 1)/2. Indeed, we have
because a is coprime to p.
This gives r = s, since r and s are positive least residues. But there are exactly (p − 1)/2 of them, so their values are a rearrangement of the integers 1, 2, …, (p − 1)/2. 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 p 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 p and q 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 1 + i in Z[i], the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.
nth power residue symbol
Let k be an algebraic number field with ring of integers
Assume that a primitive nth root of unity
This can be proved by contradiction, beginning by assuming that
and dividing by x − 1 gives
Letting x = 1 and taking residues mod
Since n and
Thus the residue classes of
There is an analogue of Fermat's theorem in
and since
is well-defined and congruent to a unique nth root of unity ζns.
This root of unity is called the nth-power residue symbol for
It can be proven that
if and only if there is an
1/n systems
Let
In other words, there are
The numbers 1, 2, … (p − 1)/2, used in the original version of the lemma, are a 1/2 system (mod p).
Constructing a 1/n system is straightforward: let M be a representative set for
The lemma for nth powers
Gauss's lemma may be extended to the nth power residue symbol as follows. Let
Then for each i, 1 ≤ i ≤ m, there are integers π(i), unique (mod m), and b(i), unique (mod n), such that
and the nth-power residue symbol is given by the formula
The classical lemma for the quadratic Legendre symbol is the special case n = 2, ζ2 = −1, A = {1, 2, …, (p − 1)/2}, b(k) = 1 if ak > p/2, b(k) = 0 if ak < p/2.
Proof
The proof of the nth-power lemma uses the same ideas that were used in the proof of the quadratic lemma.
The existence of the integers π(i) and b(i), and their uniqueness (mod m) and (mod n), respectively, come from the fact that Aμ is a representative set.
Assume that π(i) = π(j) = p, i.e.
and
Then
Because γ and
which, since A is a 1/n system, implies s = r and i = j, showing that π is a permutation of the set {1, 2, …, m}.
Then on the one hand, by the definition of the power residue symbol,
and on the other hand, since π is a permutation,
so
and since for all 1 ≤ i ≤ m, ai and
and the theorem follows from the fact that no two distinct nth roots of unity can be congruent (mod
Relation to the transfer in group theory
Let G be the multiplicative group of nonzero residue classes in Z/pZ, and let H be the subgroup {+1, −1}. Consider the following coset representatives of H in G,
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 a to (−1)n, where a and n 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.