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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 14.1  \\ Spring 2023 \hfill Due April 26}}}

\large

\vspace{1cm}


\begin{exercise}
Let $A$ and $B$ be abelian groups, let $i_A :A \to A \oplus B$ be defined by $i_A(a) = (a,0)$, and let $i_B : B \to A \oplus B$ be given by $i_B (b) = (0,b)$.  Let $\widehat{i_A} : \widehat{A \oplus B} \to \widehat{A}$ and $\widehat{i_B} : \widehat{A \oplus B} \to \widehat{B}$ be the dual maps.  Show that the map
\[
\widehat{i_A} \oplus \widehat{i_B} : \widehat{A \oplus B} \to \widehat{A} \oplus \widehat{B}
\]
defined by $\chi \mapsto \left(\widehat{i_A}(\chi), \widehat{i_B}(\chi) \right)$ is a homomorphism.
\end{exercise}


\begin{exercise}
Let $n \in \N$.  Define $f: \widehat{\Z/n \Z} \to \boldsymbol{\mu}_n$ by $f(\chi) = \chi(1)$, as in class.  Prove that $f$ is a homomorphism.
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