180 CHAPTER 16

no matter what element σ is chosen from F (p), χ

r

(σ ) will always be

the same number. The great thing is that in our theorems we will

only need to refer to this character value χ

r

(Frob

p

), and not to the

whole matrix r(Frob

p

), which is not well-deﬁned.

If p is in the set S,thenχ

r

(Frob

p

)isnot well-deﬁned, and we do

not discuss Frob

p

in relationship to r. We cannot. Frustrated, we

often refer to S as the set of “bad” primes for the representation r.

We are not being judgmental. We are just projecting our frustration

onto the primes.

In most of the examples and theorems that are important in

number theory, the set S of bad primes is ﬁnite. Usually, knowledge

of S comes if we have knowledge of how a particular representation

r is constructed. You will see examples of this later.

We should be decorating the set of bad primes S with a subscript,

such as S

r

,becauseS depends on what representation r we are

studying. But that just gives us more to worry about, and because

we will only deal with one r at a time, we just leave it out.

In summary, any Galois representation r comes with a set S of

bad primes, and if p is not in S,thenχ

r

(Frob

p

) is a dandy, well-

deﬁned, unambiguous number which we can use in our formulas .

For completeness, we are going to tell you more about how to ﬁnd

Frob

p

, but you could skim this material during your ﬁrst reading of

the book.

Algebraic Integers, Discriminants, and Norms

We start by telling you about parts of algebraic number theory that

are not as complicated as they might seem at ﬁrst:

DEFINITION: An algebraic integer is an element of Q

alg

that

is a root of a Z-polynomial that has a leading coefﬁcient

3

of 1.

The set of all algebraic integers is written

Z.Ifα is an

3

The leading coefﬁcient of a polynomial is the coefﬁcient of the highest power of x in

the polynomial. For example, the leading coefﬁcient of the polynomial 3x

5

− 2x

2

+ 7x − 1

is 3.

FROBENIUS 181

element of Z, then the Z-polynomial of smallest degree with

ﬁrst coefﬁcient 1 which has α as a root is called the minimal

polynomial for α. Note: Most algebraic integers are not

ordinary integers.

It is easy to ﬁnd an element of

Z:FindaZ-polynomial whose

ﬁrst coefﬁcient is 1. For example, any solution of x

5

+ 41x

4

+ 32x +

11 = 0 is an algebraic integer. ( Of course, we write x

5

instead of the

longer 1x

5

, but the coefﬁcient of x

5

is still 1.) It is less obvious—in

fact, rather difﬁcult to show—that sums and products of algebraic

integers are algebraic integers. Quotients of algebraic integers are

not necessarily algebraic integers. The reason for the terminology

“algebraic integer” is that these are the elements of Q

alg

that

mimic ordinary integers: They can be added and multiplied, but not

divided, without leaving the realm of algebraic integers. Of course

you can divide them, but the answer may well be only an algebraic

number, not an algebraic integer any more. Every element of Q

alg

is

a quotient of elements of Z. And one more important fact about Z:

An ordinary integer is always an algebraic integer (e.g., 137 solves

x

1

− 137 = 0),and,conversely,theonly rational numbers that are

algebraic integers are the ordinary integers. You can symbolize this

last sentence by writing

Z ∩ Q = Z.

Next, we take a bit of a detour to discuss the discriminant

of a polynomial. You may remember this term from high-school

algebra: The discriminant of the quadratic polynomial ax

2

+ bx + c

is b

2

− 4ac. You may even remember that the discriminant of a

quadratic polynomial tells you if the polynomial has 0, 1, or 2 real

roots. This, of course, is another way of saying how many points

there are on the variety S(R) associated with the polynomial. So it

is not surprising that there is a generalization of the discriminant

to higher degrees that will prove useful to us.

DEFINITION: Suppose that f (x) = x

n

+ c

n−1

x

n−1

+···+c

1

x +

c

0

is a Z-polynomial with roots α

1

, α

2

, ..., α

n

,sothatf (x) =

(x − α

1

)(x − α

2

) ···(x − α

n

). (Notice that because we are

assuming that f (x) starts with the coefﬁcient 1, all of these

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