On page 123 of Chapter 5 in Bushnell and Henniart's book The Local Langlands Conjecture for GL(2), they state an elementary property of tamely ramified extension of local fields, which is as follows,
Let $E/F$ be a tamely ramified field extension of local fields, let $e$ be the ramification index.
For $m \ge 1$, the norm map $\mathrm{Nm}_{E/F}$ induces an isomorphism $$ U_E^{em}/U_E^{em+1} \to U_F^m/U_F^{m+1} $$
On the next page they make the following definition,
Let $(E/F,\chi)$ be an admissble pair, and let n be the level of $\chi$. We say that $(E/F,\chi)$ is minimal if $\chi|_{U^n_E}$ does not factor through $\mathrm{Nm}_{E/F}$.
While on page 125 they assert to prove the following result,
Let $E/F$ be a tamely ramified quadratic field extension, and let $\chi$ be a character of $E^\times$ of level $m\ge 1$. Let $\alpha \in \mathfrak{p}_E^{-m}$ satisfy $\chi(1+x)=\chi(\alpha x)$, $x \in \mathfrak{p}^m_E$. Then $(E/F,\chi)$ is a minimal admissible pair if and only if $\alpha$ is minimal over $F$.
As we know from Chapter 4, there are indeed minimal elements that occur in the $E/F$ unramified case. Yet, if the preceding result is true, we can only get our minimal element from a minmal pair. So minimal pair shall occur in the unramified case. But according to the property stated in the beginning, if $E/F$ is unramified, and let $\chi$ be a character of $E^\times$ of level $m\ge 1$, then we will get an isomorphism induced by the norm map, say $ U_E^{m}/U_E^{m+1} \to U_F^m/U_F^{m+1} $, in this way, since the level of $\chi$ is m, the isomorphism will tell us that $\chi|_{U^m_E}$ does factor through $\mathrm{Nm}_{E/F}$. So minimal pair can never occur in the unramified case. This apparently contradicts the deduction in the beginning of this paragraph.
There must be something wrong with my understanding, could you please help me to find out which part of my thinking is wrong? Many thanks!
