Let be a prime, and let be any integer. Then we can define the Legendre symbol
We say that is a quadratic residue modulo if there exists an integer so that .
Equivalently, we can define the function as the unique nontrivial multiplicative homomorphism of into , extended by .
Quadratic Reciprocity Theorem
There are three parts. Let and be distinct odd primes. Then the following hold:
This theorem can help us evaluate Legendre symbols, since the following laws also apply:
- If , then .
There also exist quadratic reciprocity laws in other rings of integers. (I'll put that here later if I remember.)
Theorem 1. Let be an odd prime. Then .
Proof. It suffices to show that if and only if is a quadratic residue mod .
Suppose that is a quadratic residue mod . Then , for some residue mod , so by Fermat's Little Theorem.
On the other hand, suppose that . Then is even, so is an integer. Since every nonzero residue mod is a root of the polynomial and the nonzero residues cannot all be roots of the polynomial , it follows that for some residue , Therefore is a quadratic residue mod , as desired.
Now, let and be distinct odd primes, and let be the splitting field of the polynomial over the finite field . Let be a primitive th root of unity in . We define the Gaussian sum
Proof. By definition, we have Letting , we have Now, is a root of the polynomial it follows that for , while for , we have Therefore But since there are nonsquares and nonzero square mod , it follows that Therefore by Theorem 1.
Theorem 2. .
Proof. We compute the quantity in two different ways.
We first note that since in , Since , Thus
On the other hand, from the lemma, Since , we then have Since is evidently nonzero and we therefore have as desired.
Theorem 3. .
Proof. Let be the splitting field of the polynomial over ; let be a root of the polynomial in .
We note that So
On the other hand, since is a field of characteristic , Thus Now, if , then and , so , and On the other hand, if , then and , so Thus the theorem holds in all cases.
- Helmut Koch, Number Theory: Algebraic Numbers and Functions, American Mathematical Society 2000. ISBN 0-8218-2054-0