QUADRATIC RECIPROCITY 71

proof is correspondingly more complicated—we will not be giving

that proof either.

The Legendre Symbol

There is a clever bit of notation, going back to the French mathe-

matician Adrien-Marie Legendre (a contemporary of Gauss’s), that

lets us summarize what we just discovered. We will also see that

it does more than summarize; it has an important multiplicative

property (7.1) that will be interpreted in chapter 19. It is called

the Legendre symbol in his honor. Remember that we noticed that

after we select some integer a,thevarietyS(F

p

) corresponding to

the polynomial x

2

− a can have 0, 1, or 2 elements. Here is how

Legendre chose to record this fact. He deﬁned the Legendre symbol

a

p

according to the following formula:

2

a

p

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

−1 if the number of solutions to x

2

− a ≡ 0(modp)is0.

0 if the number of solutions to x

2

− a ≡ 0(modp)is1.

1 if the number of solutions to x

2

− a ≡ 0(modp)is2.

Thus, what we were exploring in the previous section was

−1

p

.

WARNING: The Legendre symbol

a

p

is not a divided by p.

Legendre used a fraction symbol but put in the parentheses to

remind you that it is not division but—the Legendre symbol!

3

2

See our previous footnote to see why S(F

p

) cannot have more than two elements. Also

note the following facts:

•

Because p is odd, b ≡ p − b (mod p) unless b ≡ 0(modp).

•

p −b ≡−b (mod p), i.e., p − b =−b as elements of the ﬁeld F

p

.

So if a is a nonzero square in F

p

, then it has two unequal square roots b and −b for some

b in F

p

.

3

If you leave out the horizontal bar, you get the symbol

a

p

, which is reserved for

something completely different called a binomial coefﬁcient.

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