O'Reilly logo

Fearless Symmetry by Robert Gross, Avner Ash

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

184 CHAPTER 16
For example, if θ is any root of x
5
+ 41x
4
+ 32x + 11 = 0, then
N(θ ) = 11.
It is a fact that N(α)N(β) = N(αβ ). The definition of norm just
given does not make this obvious, but it can be proved.
A Working Definition of Frob
p
Now that we have told you about the discriminant and the norm,
we can tell you a little bit more about our mysterious element
Frob
p
. Fix a prime p. To tell you what Frob
p
does to an element
of Q
alg
, it is good enough to tell you what Frob
p
does to every
element of
Z, because every element of Q
alg
can be written as
a quotient of elements of
Z.Inotherwords,ifα is an element
of Q
alg
, then we can write α = β/γ ,whereβ and γ are in Z.So
Frob
p
(α) = Frob
p
(β)/Frob
p
(γ ), and all we need to do is tell you about
Frob
p
(θ)whereθ equals β or γ .
So, now suppose that θ is an algebraic integer. W e have yet one
more complication: We can only define Frob
p
(θ)ifp is unramified
with respect to θ . What does this mean? Take the minimal
polynomial f of θ. Compute the discriminant
f
of this polynomial.
If p is not a factor of
f
,thenp is unramified with respect to θ.We
can only easily define Frob
p
(θ)ifp does not divide
f
.
Occasionally it can happen that p is unramified with respect to
θ even if p is a factor of
f
.Infact,whetherornotp is unramified
with respect to θ is really a property of the field Q(f ), but it is too
complicated to give the accurate definition except in an appendix to
this chapter. The simple criterion that p is not a factor of
f
will be
good enough for us. Because
f
is just a garden variety integer, it
has only finitely many prime factors. All of the other primes will be
unramified with respect to θ . (Of course, which primes those will be
depends on θ.) For example, if β so lves the fourth-degree polynomial
above, then any prime not dividing 100,368,976 will be unramified
with respect to β.
Suppose that θ is a root of the Z-polynomial f (x)withleading
coefficient 1. We know that Frob
p
(θ) has to be one of the numbers
that solves the equation f (x) = 0. Which ones can it be?

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required