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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).
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.)