We begin to explore Z-equations by looking very closely at

a fairly simple example: x

2

= a,wherea is some constant

integer. It is hard to get simpler than this equation and

still have something interesting. In this case, the interest

lies not in solving the equation in Z, R,orC, all of which

are easy, but in F

p

for various prime numbers p.

Quadratic reciprocity refers to a mysterious relation-

ship between the solutions to x

2

= q in F

p

and x

2

= p in

F

q

,wherep and q are two different odd prime numbers.

Quadratic reciprocity was the ﬁrst of all reciprocity laws,

and it is closely connected with the theory of Galois

representations. We still have not deﬁned these things,

but we want to have the example of quadratic reciprocity

under our belt before we do.

Quadratic reciprocity is part of “classical” number

theory going back to the eighteenth century. It inspired

the many other reciprocity laws that eventually developed

into a large part of modern number theory.

The Simplest Polynomial Equations

Let us discuss the solutions of polynomial equations. Since the

time of the French mathematician and philosopher Ren

´

e Descartes,

mathematicians have realized that the degree of a polynomial is

a good indication of how complicated it is. Recall that the degree

68 CHAPTER 7

of a polynomial f (x) of one variable is the highest power of x that

appears. For example, 10x

33

− 100x

7

+ 1,729 has degree 33.

A polynomial of degree 0 is simply a constant. Because a rose is

a rose is a rose, there is not much to say about a single constant,

and we can go on to the next case.

A polynomial of degree 1 is, for example, 5x + 3. The set of

solutions is again completely understandable. In any ﬁeld at all

(excluding those where we are not allowed to divide by 5, such

as F

5

), this polynomial has exactly one root, which we can ﬁnd by

dividing −3by5.

So the ﬁrst nontrivial polynomials are those of degree 2. We will

take this chapter to study just one aspect of quadratic equations. As

you will see, these matters are truly not trivial, and lead to some

very pretty and historically crucial bits of mathematics. In fact, the

subject matter here, quadratic reciprocity, is the tip of the gigantic

iceberg of “reciprocity laws” that we will explain later.

Rather than taking the most complicated degree 2 equation,

which would look like Ax

2

+ Bx + C = 0, choose just one integer, call

it a, and look at the equation x

2

− a.Ifa is 0, then the solution

set of x

2

− 0 = 0 will just be x = 0, and so it will contain one num-

ber. Otherwise, the variety deﬁned by x

2

− a will usually contain

either two numbers or none, depending on whether or not a is a

square.

For example, 1 is 1

2

in every number system. If we take a = 1,

and look at x

2

− 1, then S(Z) ={1, −1},andS(Q), S(R), and S(C)

will be the same set. If we take x

2

+ 1, for instance (meaning that

a =−1; we never said that a had to be positive), then S(Z)hasno

elements, and S(Q) has no elements, and S(R) has no elements, and

S(C) ={i, −i}. What happened here is that −1isasquareinC but

not in Z, Q,orR.

1

We are going to play this game now using F

p

for various primes

p, after ﬁxing some integer a. It is not so interesting to let p = 2:

if a = 0, then S(F

2

) ={0},andifa = 1, then S(F

2

) ={1}. Therefore,

1

In general, if R is a ﬁeld and a = b

2

with a and b elements of R,thenx

2

− a =

(x − b)(x + b) equals 0 exactly if x = b or x =−b,andthenS(R) ={b, −b}. There will be

two different elements in S(R) unless b =−b, which only happens if R has characteristic

2orifb = 0. If a is not a square in R,thenx

2

− a = 0 h as no solutions and S(R)isempty.

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