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\hfill \textbf{February 27, 2015}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 2\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}
\begin{enumerate}
\item
Show that the class $\mathcal{F}$ of fields of characteristic $0$ does not admit sums or products. 
\item
Show that the class $\mathcal{T}$ of all
abelian torsion groups admits products as
well as  sums.
\item
Let $G=<x>$ be a cyclic group of order $n$. Find a condition on $r,s,n$ that is equivalent to $<x^r>\subseteq<x^s>$.
\item
Let $p(x)$ be any polynomial in $\mbb{F}[x]$ of degree $n$. Find an $n \times n-$ matrix $A$ such that $p(A)=0$. 
\item  Let $\mbf{A}=\mbb{Z}\oplus \mbb{Z}$ be
the free $\mbb{Z}-$module of rank 2. Let $\Delta=\{(n,n)\mid\in
\mbb{Z}\}$. Show that $\Delta$ is a submodule as well as a
homomorphic image of $\mbf{A}$. Is $\Delta$ adirect summand of
$\mbf{A}$?
\item If $\mbf{A}$ is
the free abelian group, freely generated by a countably infinite
set, then one has that $\mbf{A}\oplus\mbf{Z}\oplus
\mbf{Z}\iso\mbf{A}\oplus\mbb{Z}$ but $\mbb{Z}\oplus \mbb{Z}$ and
$\mbb{Z}$ are not isomorphic.
\end{enumerate} \end{document}

$(n,m)=(n,n)+(m-n)(0,1)$

\item
A linear operator $T:\mathbb{R}^n\to \mathbb{R}^n$ is symmetric if $(Tv,w)=(v,Tw)$ holds for all $v$ and $w$ in $\mathbb{R}^n$. Show that the minimal polynomial $p_T(x)$ of $T$ is a product
 of $k-$many linear factors $(x-\lambda)$ with real eigenvalues $\lambda$ and that $\mathbb{R}^n$ decomposes into a direct sum of $k-$many eigen-spaces which are pairwise perpendicular. Each eigen-space has of course an orthogonal basis.
\item Find $\mbb{Z}/(36)(3)$.
