Quadratic Residues

Theorem 3.1 (page 132)

Let p be an odd prime. Then

1. ,

2. ,

3.

4. If (a,p) = 1, then

5.

Proof. 1 and 5 were proved on Monday.

2. Since and we have .

3. Use part 1 again.

4. since a² is a square. Now apply part 2.

We state the next two theorems without proof.

Theorem 3.3 For any odd prime p, .

Theorem 3.4 (page 137)

Gauss’ Law of Quadratic Reciprocity

If p and q are distinct odd primes, then

These theorems allow us to answer the question “Is a a quadratic residue mod p?” rather easily, without large computations.

Example. Is 7 a quadratic residue mod 59?

Compute . By Quadratic Reciprocity we have . By reducing mod 7 we have

. We apply reciprocity to .

and

Thus .

.

So . 7 is a quadratic residue mod 59.

Example. Is 30 a square mod 89?

.

. .

Thus

.

Thus we have .

30 is not a perfect square mod 89.

Now a harder problem.

Find all odd primes, p, such that .

Let p be an odd prime, different from 3. Then.

Since p is an odd prime, different from 3 we have that .

Case 1: .

In this case . Then if (p-1)/2 is an even number. This is true if (p-1)/2 = 2k for some integer k. This reduces to p = 4k+1 or

We now have two congruences to solve simultaneously (using the Chinese Remainder Theorem):

The solution is

Case 2:

In this case we have shown that .

reduces to =. We now need to have (p-1)/2 be an odd number. If (p-1)/2 = 2k+1 then p=4k+3 or

The two congruences to solve in this case are:

The only solution is

Our final solution is 3 is a quadratic residue mod p for p an odd prime if and only if p is congruent to either 1 or 11 mod 12.