FROBENIUS 185

FACT: Frob

p

(θ) solves the minimal polynomial for θ, and has

the lovely property that the ordinary integer N(Frob

p

(θ) − θ

p

)

is evenly divisible by p. That is, it leaves no remainder when

divided by p. In particular, we are telling you that there is at

least one number—call it β—that both solves the minimal

polynomial for θ and makes N(β − θ

p

) evenly divisible by p.

In case there is only one such β, we deﬁne Frob

p

(θ)tobethis

number β. (It is possible that sometimes β will equal θ itself .)

The deﬁnition of Frob

p

(θ) when there are several β’s to choose

from is given in the second appendix to this chapter.

An Example of Computing Frobenius Elements

An example, using the easiest possible nonrational element of Z,

is Frob

p

(i). We cannot deﬁne Frob

2

(i), because 2 is ramiﬁed with

respect to i: The minimal Z-equation that i satisﬁes is x

2

+ 1 = 0.

We know from the formula for the discriminant of a quadratic

polynomial that this has discriminant −4. Therefore,

f

is evenly

divisible by 2, and in fact 2 is ramiﬁed with respect to i,sowe

cannot deﬁne Frob

2

(i).

What about Frob

3

(i)? We need to start with the lowest-degree

equation that i solves, which is x

2

+ 1 = 0. The other solution of

that equation is −i.SoFrob

3

(i)—indeed Frob

p

(i) for any odd prime

p—has to be either i or −i. We also need to have

3|N(Frob

3

(i) − i

3

).

Here we are using the handy notation p|a to mean “a is evenly

divisible by p.” Because i

3

=−i, N(Frob

3

(i) + i) has to be a multiple

of 3. If we tried Frob

3

(i) = i,wewouldgetN(2i), which is not

a multiple of 3. (Why not? The minimal Z-polynomial for 2i is

x

2

+ 4. It has constant term 4. Therefore, N(2i) = 4, and 4 is not

evenly divisible by 3.) So we need to have Frob

3

(i) =−i.(Check:

N(−i + i) = N(0) = 0, which is evenly divisible by 3.)

How about Frob

5

(i)? Now we get

5|N(Frob

5

(i) − i

5

)

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