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\begin{flushright}
  Your Name Here
\end{flushright}


\begin{center}
  \textbf{Math 105}\\ Homework 6
\end{center}


For a local field $E$ and $\alpha\in E^\times$, define $\ord_E(\alpha)
= m$ where $\alpha = \pi_E^m u$, $u \in U_E=O_E^\times$.

\begin{enumerate}

  
\item (4-3-18) Let $E/F$ be an extension of local fields,
  $|\alpha|_E = |N_{E/F}(\alpha)|_F^{1/[E:F]}$.  Show that for
  $\alpha\in E^\times$,
  $\ord_F(N_{E/F}(\alpha)) = f(E/F) \ord_E(\alpha)$.

\item (4-3-18) Let $E/F$ be an extension of local fields. For
  $\alpha \in E^\times$, let $I = \alpha\O_E$ and define:
  $N_{E/F}(I) := N_{E/F}(\alpha)\O_F$.  Show that this definition is
  independent of the choice of generator, and induces a homomorphism
  from the group of $E$-ideals to the group of $F$-ideals.  Finally,
  show that $N_{E/F}(\sP_E) = \sP_F^f(E/F)$.
\item (4-3-18) Let $E$ be a local field.  Show that $E$ contains the
  $(N\sP_E-1)$st roots of unity.

\item (4-3-18) Let $E$ be a local field.  The goal is to show that
  $E^\times \cong \la\pi_E\ra \times U_E^0\times (1+\sP_E)$ where
  $U_E^0$ is the group of $N\sP_E-1$st roots of unity in $E$, and
  $(1+\sP_E)$ is the subgroup of `principal' units of $E$.  Start by
  showing that $U_E^0$ and $(1+\sP_E)$ are subgroups of
  $U_E=\O_E^\times$ with $U_E= U_E^0\times (1+\sP_E)$.
  

     
\end{enumerate}

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