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188 CHAPTER 16
For instance, consider f (x) = x
2
11. The discriminant of f is 44.
So consider some prime p different from 2 or 11. We use Theorem
16.1 to compute Frob
p
(
11). (Remember that
11 is by deﬁnition
the positive square root, which is approximately 3.3166247904.)
We know Frob
p
on the set of roots {
11,
11} is either the
identity permutation (the neutral element in
{
11,
11}
)orthe
permutation that switches the two roots. In the ﬁrst case, Frob
p
has two cycles each of length 1, and in the second case, it has only
one cycle of length 2.
Now f (x) factors in F
p
into two simpler factors if and only if 11
is a square modulo p.
5
So, by Theorem 16.1, the cycle type of Frob
p
is 1 + 1 if 11 is a square modulo p, and is 2 otherwise. In the ﬁrst
case, Frob
p
(
11) =
11 and in the second Frob
p
(
11) =−
11. For
example, Frob
89
(
11) =
11 because x
2
11 factors into two linear
factors in F
89
.
APPENDIX
The Ofﬁcial Deﬁnition of the Bad Primes
for a Galois Representation
Suppose we have a Galois representation r. Remember that this
means r is a morphism of G
Q
, the absolute Galois group of Q (which
we are abbreviating G) to some other group H,eitheragroupof
permutations or a group of matrices.
Any morphism f from a group G to a group H has what is called
a kernel. This is just the set of elements x in G with the property
that f (x) is the neutral element e of H. We call the kernel of r by
the name K
r
. Because ﬁnding the kernel is the same as solving the
equation f (x) = e, you can imagine that this is an important set to
study. For example, r is a faithful representation precisely when K
r
contains only the neutral element e.
So K
r
is a subset of G. It actually is a group all by itself, but that
is not so important right now.
5
That is, if and only if 11 = a
2
for some a in F
p
.

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