Proof.
Let
![$ \tilde{a} \in \mathbf{F}_\mathfrak{p}$](img1340.png)
be an element such that
![$ \mathbf{F}_\mathfrak{p}=\mathbf{F}_p(a)$](img1334.png)
.
Lift
![$ \tilde{a}$](img1341.png)
to an algebraic integer
![$ a\in\O _K$](img494.png)
, and let
![$ f=\prod_{\sigma\in D_p}(x-\sigma(a))\in K^D[x]$](img1342.png)
be the characteristic polynomial of
![$ a$](img163.png)
over
![$ K^D$](img1293.png)
.
Using Proposition
14.1.4 we see that
![$ f$](img162.png)
reduces to the minimal polynomial
![$ \tilde{f}=\prod (x-\tilde{\sigma(a)})\in \mathbf{F}_p[x]$](img1343.png)
of
![$ \tilde{a}$](img1341.png)
(by the Proposition the coefficients of
![$ \tilde{f}$](img1344.png)
are in
![$ \mathbf{F}_p$](img549.png)
, and
![$ \tilde{a}$](img1341.png)
satisfies
![$ \tilde{f}$](img1344.png)
, and the
degree of
![$ \tilde{f}$](img1344.png)
equals the degree of the minimal polynomial
of
![$ \tilde{a}$](img1341.png)
). The roots of
![$ \tilde{f}$](img1344.png)
are of the form
![$ \tilde{\sigma}(a)$](img1345.png)
, and
the element
![$ \Frob _p(a)$](img1346.png)
is also a root of
![$ \tilde{f}$](img1344.png)
, so it is of the form
![$ \tilde{\sigma(a)}$](img1347.png)
.
We conclude that the generator
![$ \Frob _p$](img1327.png)
of
![$ {\mathrm{Gal}}(\mathbf{F}_\mathfrak{p}/\mathbf{F}_p)$](img1324.png)
is
in the image of
![$ \varphi $](img52.png)
, which proves the theorem.
Proof.
By definition
for all ![$ a\in\O _K\}$](img1357.png)
,
so it suffices to show that if
![$ \sigma\not\in D_\mathfrak{p}$](img1358.png)
, then there
exists
![$ a\in\O _K$](img494.png)
such that
![$ \sigma(a)=a\pmod{\mathfrak{p}}$](img1359.png)
.
If
![$ \sigma\not\in D_\mathfrak{p}$](img1358.png)
, we have
![$ \sigma^{-1}(\mathfrak{p})\neq \mathfrak{p}$](img1360.png)
, so
since both are maximal ideals, there exists
![$ a\in\mathfrak{p}$](img1189.png)
with
![$ a\not\in\sigma^{-1}(\mathfrak{p})$](img1361.png)
,
i.e.,
![$ \sigma(a)\not\in\mathfrak{p}$](img1362.png)
. Thus
![$ \sigma(a)\not\equiv a\pmod{\mathfrak{p}}$](img1363.png)
.