Next we ﬁnd all the algebraic numbers α with the property that
they are not moved by any element in K
be the set
of all α in Q
such that σ (α) = α for every σ in K
. It turns out E
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
, 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
deﬁnition of this last phrase is given in the next appendix.
The Ofﬁcial Deﬁnition of “Unramiﬁed” and Frob
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
, is deﬁned to be K ∩ Z.
DEFINITION: An ideal of O
is a subset A of O
under addition, subtraction, and multiplication, containing 0
and satisfying the property that xa is in A whenever x is in
and a is in A.
DEFINITION: If A and B are two ideals, the product ideal
AB is the set of all sums a
, ..., a
are in A and b
, ..., b
are in B.Ifk is a positive integer, A
means the product of A with itself k times.