FROBENIUS 189

Next we ﬁnd all the algebraic numbers α with the property that

they are not moved by any element in K

r

.Thatis,letE

r

be the set

of all α in Q

alg

such that σ (α) = α for every σ in K

r

. It turns out E

r

is actually a ﬁeld all by itself, but that is not so important right

now either. So given any Galois representation r, we get a set of

algebraic numbers E

r

, which is called the ﬁxed ﬁeld of the kernel

of r. Here is the payoff: The prime p is unramiﬁed with respect to

r if and only if it is unramiﬁed for every α contained in E

r

.The

deﬁnition of this last phrase is given in the next appendix.

APPENDIX

The Ofﬁcial Deﬁnition of “Unramiﬁed” and Frob

p

In general, if K = Q(f )forsomeZ-polynomial f , we now deﬁne

what it means for p to be “unramiﬁed in K.” If f is the minimal

polynomial for θ,wealsosayp is unramiﬁed for θ under the same

circumstances. Then we describe the set of images of Frobenius

elements at p under the restriction morphism from G to G(f ). We

just present the deﬁnitions without explanation. As we said, full

explanation would require turning this appendix into a textbook on

algebraic number theory.

•

DEFINITION: The set of algebraic integers in K, written as

O

K

, is deﬁned to be K ∩ Z.

•

DEFINITION: An ideal of O

K

is a subset A of O

K

closed

under addition, subtraction, and multiplication, containing 0

and satisfying the property that xa is in A whenever x is in

O

K

and a is in A.

•

DEFINITION: If A and B are two ideals, the product ideal

AB is the set of all sums a

1

b

1

+···+a

m

b

m

,wherea

1

, ..., a

m

are in A and b

1

, ..., b

m

are in B.Ifk is a positive integer, A

k

means the product of A with itself k times.

190 CHAPTER 16

•

DEFINITION: An ideal A is called a prime ideal if the

following property holds: If x and y are in O

K

and xy is in A,

then either x or y or both must have already been in A.

•

DEFINITION: If d is any element of O

K

,(d)isthesetofall

products xd where x is in O

K

.Notethat(d) is an ideal.

•

THEOREM: For any prime integer p in Z, there are prime

ideals P

1

, ..., P

t

in O

K

and a positive integer e such that

(p) = P

e

1

P

e

2

···P

e

t

.

•

DEFINITION: The prime number p is unramiﬁed in K if

and only if e = 1.

•

THEOREM: Let p and its ideal factorization be given as

above. Suppose that p is unramiﬁed in K.Letj be an integer

from 1 to t. Then there exists exactly one element σ in the

Galois group G(f ) with the following two properties:

1. For any x in P

j

, σ (x) is again in P

j

.

2. For any y in O

K

, σ (y) − y

p

is in P

j

.

•

THEOREM: The element σ is one possibility for r

G( f )

(Frob

p

)

in G(f ). You get all the possibilities by varying j.

•

THEOREM: For any β in G(f ), there are inﬁnitely many

primes q such that β is a possible r

G( f )

(Frob

q

). Thus, from the

point of view of the ﬁnite Galois group G(f ), being a Frobenius

element is not unusual—quite the opposite. What is difﬁcult

is knowing which q’s go with which conjugacy classes in G(f ).

That is one thing reciprocity laws are meant to help us with.

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