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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