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\usepackage{ulem}
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\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \#11}
\duedate{10/14/10}

\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote (7.3) 10, 13, 26, 29}

%remember, $R$ is a ring with identity $1 \neq 0$.
\begin{problem}[7.3.10]
Decide which of the following are ideals of the ring $\Z[x]$:
\begin{itemize}
\item[(a)] the set of all polynomials whose constant term is a multiple of 3
\item[(b)] the set of all polynomials whose coefficient of $x^2$ is a multiple of 3
\item[(c)] the set of all polynomials whose constant term, coefficient of $x$ and coefficient of $x^2$ are zero
\item[(d)] $\Z[x^2]$ (i.e., the polynomials in which only even powers of $x$ appear)
\item[(e)] the set of polynomials whose coefficients sum to zero
\item[(f)] the set of polynomials $p(x)$ such that $p'(0)=0$, where $p'(x)$ is the usual first derivative of $p(x)$ with respect to $x$.
\end{itemize}

\end{problem}
\newpage

\begin{problem}[7.3.13]
Prove that the ring $M_2(\R)$ contains a subring that is isomorphic to $\C$.
\end{problem}
\newpage

\begin{problem}[7.3.26]
The {\it characteristic} of a ring $R$ is the smallest positive integer $n$ such that \\ $1+1+\dots+1=0$ ($n$ times) in $R$; if no such integer exists the characteristic of $R$ is said to be 0.  For example, $\Z/n\Z$ is a ring of characteristic $n$ for each positive integer $n$ and $\Z$ is a ring of characteristic 0.
\begin{itemize}
\item[(a)] Prove that the map $\Z \rightarrow R$ defined by
$$k \mapsto \begin{cases} 
1+1+\dots+1 \ (k \text{ times}) &\text{if } k > 0 \\
0 &\text{if } k = 0 \\
-1-1-\dots-1 \ (k \text{ times}) &\text{if } k < 0
\end{cases}$$
is a ring homomorphism whose kernel is $n\Z$, where $n$ is the characteristic of $R$ (this explains the use of the terminology ``characteristic 0'' instead of the archaic phrase ``characteristic $\infty$'' for rings in which no sum of 1's is zero).
\item[(b)] Determine the characteristics of the rings $\Q$, $\Z[x]$, $\Z/n\Z[x]$.
\item[(c)] Prove that if $p$ is a prime and if $R$ is a commutative ring of characteristic $p$, then $(a+b)^p = a^p + b^p$ for all $a, b \in R$.
\end{itemize}

\end{problem}
\newpage

\begin{problem}[7.3.29]
Let $R$ be a commutative ring.  Recall (cf. Exercise 13, Section 1) that an element $x \in R$ is nilpotent if $x^n=0$ for some $n \in \Z^+$.  Prove that the set of nilpotent elements form an ideal --- called the {\it nilradical} of $R$ and denoted by $\mathfrak{N}(R)$.  [Use the Binomial Theorem to show $\mathfrak{N}(R)$ is closed under addition.]
\end{problem}

\end{document}