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?

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