RECIPROCITY LAWS 197

by means of the Frobenius elements, will match the output from

one of a certain set of black boxes (those with the correct label).

Sometimes, reciprocity laws are theorems that have been proven.

Sometimes they are only conjectured and have not been proven yet.

And, of course, you can mistakenly conjecture a false reciprocity

law and ﬁnd a numerical disproof. But the reciprocity laws in the

remainder of this book are all either theorems or conjectures that

most experts would say are probably going to turn out to become

theorems.

Digression: Conjecture

The word conjecture means “guess.” The way it is used in math-

ematics is “educated guess.” In this digression, we will mention

several conjectures from parts of mathematics other than Galois

theory.

There are at least two different classes of great conjectures:

1. Those based on evidence.

2. Those based on analogy.

Some conjectures that are currently the object of intense study by

mathematicians include the Poincar

´

e Conjecture (which reportedly

may have been proven true as of this writing), the Riemann

Hypothesis (which is a conjecture, though it is called a “hypothe-

sis”), and the conjecture that “P = NP.” As a matter of fact, there is

a million-dollar prize (offered by the Clay Mathematics Institute)

for the ﬁrst resolution of any of these three conjectures, as well as

some others.

2

Mathematicians may not increase their efforts be-

cause of the prize, but public interest has certainly increased.

Sometimes, serious conjectures are proven false. An example is

the Mertens Conjecture. First deﬁne the M

¨

obius function µ(n) for

positive integers n: µ(n) = 0ifn is divisible by a square number

larger than 1; otherwise, µ(n) = 1ifn has an even number of prime

factors and µ(n) =−1ifn has an odd number of prime factors. Then

2

A discussion of the Millennium problems may be found in (Devlin, 2002).

198 CHAPTER 17

you deﬁne the Mertens function for x > 1by

M(x) = µ(1) + µ(2) + µ(3) +···+µ(n),

where n is the largest integer less than or equal to x. The Mertens

Conjecture states that for all x > 1,

|M(x)| <

√

x.

There is a known way to prove the Riemann Hypothesis from this

statement, if it were true. If you work out M(x) for small values of x,

the conjecture seems to be true. In 1985, however, Andrew Odlyzko

and Herman te Riele proved that the Mertens Conjecture is false.

Fermat’s Last Theorem states: If x, y, and z are all nonzero

integers, and n is an integer greater than 2, then it cannot happen

that x

n

+ y

n

= z

n

. Before this theorem was proved by Andrew Wiles,

it was a conjecture that fell in both classes, which is why it

was regarded as particularly strong. In the simplest sense, there

was evidence supporting it, as no one ever found integers solving

x

n

+ y

n

= z

n

. The supporting evidence in fact was much stronger:

For many particular values of n, mathematicians, starting in the

eighteenth century, had proved that there were no solutions of

x

n

+ y

n

= z

n

.

There was also at least one analogy that helped convince many

mathematicians that Fermat’s Last Theorem was true. There are

many statements about integers that can also be made about

polynomials, and in general the true statements about integers

tend also to be true of polynomials. Proving things about polyno-

mials is usually much simpler than proving things about integers,

for at least two reasons: Polynomials have roots, and they can

be differentiated. Using these tools, it is not all that difﬁcult to

show that there are no nonconstant polynomials with complex

coefﬁcients f (x), g(x), and h(x) so that f (x)

n

+ g(x)

n

= h(x)

n

if n > 2.

Another conjecture that is strongly supported by evidence in both

categories is called the Riemann Hypothesis. This is the statement

that if the real part of the complex number s is positive, then the

Riemann ζ -function ζ (s) is zero only when s =

1

2

+ it (where t is a

real number). On the one hand, you can do numerical computations

and the Riemann Hypothesis looks true. On the other hand, there

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