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Fearless Symmetry by Robert Gross, Avner Ash

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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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