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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 5.1  \\ Spring 2019 \hfill Due February 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $G$ be the (additive) group of polynomials with real coefficients.  For each $f \in G$, let $\int f$ denote the antiderivative of $f$ whose graph passes through the point $(0,0)$.  Show that the mapping $f \mapsto \int f$ from $G$ to $G$ is a homomorphism.  What is its kernel?  Can $(0,0)$ be replaced by any other point in $\R^2$?
\end{exercise}

\begin{exercise}
Define $f : D_4 \to \{ \pm 1 \}$ by
\[
f(x) = \begin{cases} 1 & \text{ if } x \text{ is a rotation,} \\ -1 & \text{ otherwise.} \end{cases}
\]
Prove that $f$ is a homomorphism.  What is its kernel?
\end{exercise}

\begin{exercise}
Let $A$ be an (additive) abelian group and $B, C < A$.  Define $f: B \times C \to A$ by $f(b,c) = b+c$.

\begin{itemize}

\item[\bf a.]  Prove that $f$ is a homomorphism.

\item[\bf b.]  Use part {\bf{a}} to prove that $B + C = \{b + c \, | \, b \in B, c \in C \}$ is a subgroup of $A$.

\item[\bf c.]  Compute $\operatorname{ker} f$.

\end{itemize}

\end{exercise}


%\begin{exercise}
%Let $A$ be an abelian group, written additively.

%\begin{itemize}

%\item[\bf a.]  Let $a,b \in A$ and $n \in \Z$.  Prove that $n(a+b) = na + nb$.  [{\em Suggestion:}  Treat the cases $n>0$, $n=0$ and $n<0$ separately.  Induction may be helpful.]

%\item[\bf b.]  For $n \in \N$, let  $A[n] = \{ a \in A \, | \, na = 0 \}$.  Prove that $A[n] < A$.  $A[n]$ is called the {\em $n$-torsion subgroup} of $A$.

%\item[\bf c.] Let
%\[
%A_{\text{tors}} = \bigcup_{n \in \N} A[n].
%\]
%Prove that $A_{\text{tors}} < A$.  $A_{\text{tors}}$ is called the {\em torsion subgroup} of $A$.

%\end{itemize}

%\end{exercise}

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