Proof.
Let
be an element such that
.
Lift
to an algebraic integer
, and let
be the characteristic polynomial of
over
.
Using Proposition
9.3.4 we see that
reduces to a multiple of the minimal polynomial
of
(by the Proposition the coefficients of
are in
, and
satisfies
).
The roots of
are of the form
, and
the element
is also a root of
, so it is of the form
.
We conclude that the generator
of
is
in the image of
, which proves the theorem.
Proof.
By definition
for all , so it suffices to show that
if
, then there exists
such that
. If
, then
, so
. Since both
are maximal ideals, there exists
with
, i.e.,
. Thus
.