186 CHAPTER 16

which says that 5|N(Frob

5

(i) − i). If we try Frob

5

(i) =−i,we

get 5|N(−2i), which is not true, so the only possibility is that

Frob

5

(i) = i.

EXERCISE: Let p be an odd prime. Show that

Frob

p

(i) =

i if p ≡ 1(mod4)

−i if p ≡ 3(mod4).

Notice that this formula looks an awful lot like equation (7.4) on

page 79. This is not a coincidence. See chapter 19.

Unfortunately, this example is misleading, for in the general case

there can be more than one root of the minimal polynomial that

satisﬁes the divisibility relationship. For the complete deﬁnition of

Frob

p

(θ), see the second appendix to this chapter.

We have been discussing ramiﬁcation with respect to θ ,and

at the beginning of this chapter we spoke of ramiﬁcation with

respect to a Galois representation r. What is the connection? Well,

we would have to spell out the connection between number ﬁelds

and Galois representations, which we do in the ﬁrst appendix to

this chapter.

Frob

p

and Factoring Polynomials modulo p

Now that we have told you what Frob

p

looks like, at least some of

the time, we should tell you something about it. A lot of the amazing

things will wait for future chapters, but here is one thing that we

can use to ﬁnish off this chapter.

Pick an irreducible Z-polynomial f (x)ofdegreen with leading

coefﬁcient 1. (Irreducible just means that it does not factor into

Z-polynomials of smaller degree. A consequence of irreducibility

is that f (x) is the minimal polynomial of each of its roots.) We

can take the n roots α

1

, α

2

, ... , α

n

of this polynomial. Then we

know that any element of the Galois group G permutes these roots.

In particular, pick any prime p so that

f

is not evenly divisible

by p.Thenp is unramiﬁed with respect to any of the roots of f (x).

FROBENIUS 187

We know that Frob

p

permutes the roots. What can we say about this

permutation?

As we have seen, permutations can be broken up into cycles.We

can start with a root α

1

, apply the permutation Frob

p

to it to get

another root, apply Frob

p

to that root, and keep going. Eventually,

we have to get back to α

1

, because there are only ﬁnitely many roots.

The number of different roots that we visit on our trip is called the

length of a cycle. For example, if Frob

3

(α

1

) = α

2

,andFrob

3

(α

2

) = α

11

and Frob

3

(α

11

) = α

8

,andFrob

3

(α

8

) = α

1

, then this cycle has length 4,

because it visits the four different roots α

1

, α

2

, α

11

,andα

8

.

There will also be another cycle, starting at α

3

,becauseα

3

was

not part of the previous cycle. We can keep going, putting each root

of the polynomial into a cycle, and counting the lengths of the cycles .

The lengths have to add up to n, because each root is in exactly one

cycle. (If you are worried about the possibility that Frob

3

(α

5

) = α

5

,

we call that a cycle of length 1.) So we have a bunch of positive

integers n

1

, n

2

, ..., n

k

,sothatn

1

+ n

2

+···+n

k

= n.Theseintegers

are the lengths of the cycles produced by the permutation on the

roots. We say that Frob

p

“has cycle type n

1

+ n

2

+···+n

k

.”

4

We can also try factoring the polynomial f (x)inF

p

. For exam-

ple, x

2

− 11 = (x + 10)(x − 10) in F

89

. Here is an amazing theorem

connecting cycle types and factorizations:

THEOREM 16.1: If f (x) is an irreducible Z-polynomial and p

is a prime not dividing

f

,andiff (x) factors in F

p

into k

factors, and the degrees of those k factors are n

1

, n

2

, ..., n

k

,

then the cycle type of Frob

p

is n

1

+ n

2

+···+n

k

.

In general, it is easy to factor polynomials with elements in F

p

(there are fast computer algorithms for this), but it is difﬁcult to

ﬁgure out Frob

p

. So we usually use the factorization to tell us about

the cycle type of Frob

p

.

4

You may object that we are ignoring the ambiguity in the notation Frob

p

here. However,

the cycle type will turn out to be the same, whatever choice of Frob

p

you make. Also,

remember that the “+” sign in the cycle type does not denote addition, but is the

traditional symbol for separating the lengths of the cycles from each other.

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