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\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#1}
\date{7 January 2015; due Wednesday, 14 January 2015}

\maketitle

\begin{problab}{1.1}
Show that the polynomial $f(x)=x^3-x^2+1$ is irreducible over $\Q$, and express in radicals a solution to the equation $f(x)=0$.
\end{problab}

%\begin{solu}
%\end{solu}

\begin{problab}{1.2 (M1-1)}
Let $F=\Q(\alpha)$ where $\alpha^3-\alpha^2+\alpha+2=0$.  Express $(\alpha^2+\alpha+1)(\alpha^2-\alpha)$ and $(\alpha-1)^{-1}$ in the form $a\alpha^2+b\alpha+c$ with $a,b,c \in \Q$.
\end{problab}

\begin{problab}{1.3}
Let $D \in \Z \setminus \{0\}$ be squarefree and let $F=\Q(\sqrt{D})$.  Let $\alpha=a+b\sqrt{D} \in F$.  
\begin{enumalph}
\item Show that the ``multiplication by $\alpha$'' map 
\begin{align*}
\phi:F &\to F \\
\beta &\mapsto \phi(\beta)=\alpha\beta
\end{align*}
is a linear transformation of vector spaces over $\Q$.
\item Compute the matrix of $\phi$ on the basis $1,\sqrt{D}$ of $F$.  
\end{enumalph}
\end{problab}

\begin{problab}{1.4 (sorta M1-3)}
Let $F$ be a field and $f(X) \in F[X]$.
\begin{enumalph}
\item Show that $f(X)$ can have at most $\deg f$ roots.
\item Let $G$ be a finite abelian group.  Show that $G$ is cyclic if and only if $G$ has at most $m$ elements of order dividing $m$ for each divisor $m$ of $\#G$.
\item Deduce that if $H \subseteq F^\times$ is a finite subgroup, then $H$ is cyclic.
\end{enumalph}
\end{problab}

\begin{problab}{1.5 (sorta M1-2)}
Let $F=\Q(\sqrt{2},\sqrt{3})$.  
\begin{enumalph}
\item Determine $[F:\Q]$.  
\item Find an element $\alpha \in F$ such that $F=\Q(\alpha)$ and compute its minimal polynomial.
\end{enumalph}
\end{problab}

\begin{problab}{1.6}
Let $F$ be a field.  Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$.
\end{problab}

\end{document}