184 CHAPTER 16
For example, if θ is any root of x
+ 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
Now that we have told you about the discriminant and the norm,
we can tell you a little bit more about our mysterious element
. Fix a prime p. To tell you what Frob
does to an element
, it is good enough to tell you what Frob
does to every
Z, because every element of Q
can be written as
a quotient of elements of
Z.Inotherwords,ifα is an element
, then we can write α = β/γ ,whereβ and γ are in Z.So
(α) = Frob
(γ ), and all we need to do is tell you about
(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
(θ)ifp is unramiﬁed
with respect to θ . What does this mean? Take the minimal
polynomial f of θ. Compute the discriminant
of this polynomial.
If p is not a factor of
,thenp is unramiﬁed with respect to θ.We
can only easily deﬁne Frob
(θ)ifp does not divide
Occasionally it can happen that p is unramiﬁed with respect to
θ even if p is a factor of
.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
good enough for us. Because
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
(θ) has to be one of the numbers
that solves the equation f (x) = 0. Which ones can it be?