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Fearless Symmetry by Robert Gross, Avner Ash

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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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