\documentclass[11pt]{article}
\usepackage{amsmath}
\pagestyle{empty}
\newcommand{\sig}{\sigma}
\newcommand{\ra}{\rightarrow}
\newcommand{\eps}{\epsilon}
\newcommand{\ah}{\alpha}
\newcommand{\lam}{\lambda}
\newcommand{\gam}{\gamma}
\newcommand{\ol}{\overline}
\newcommand{\noi}{\noindent}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\beq}{\begin{quote}}
\newcommand{\enq}{\end{quote}}
\newcommand{\hsone}{\hspace*{1cm}}
\newcommand{\hstwo}{\hspace*{2cm}}
\newcommand{\So}{\\ {\tt Solution:}}
\begin{document}
\begin{center} {\Large\bf Algebra V63.0349 \\ Assignment 5} \\ Due,
Friday, Feb 26 in Recitation \end{center}
\beq
May my mind stroll about hungry and fearless \\ \hspace*{1cm} and thirsty and supple
\\ and even if it's Sunday may I be wrong
\\ for whenever men are right, they are not young
\\ -- e.e. cummings
\enq
\ben
\item Let $a,b,c,d$ be real numbers.
Let $\ah=a+bi\in C$. Let $\beta=c+di\in C$ be such that $\beta^2=\ah$.
(That is, $\beta$ is one of the two square roots of $\ah$.) Let $K$ be a field
with $Q\subseteq K$ and $a,b\in K$. Find an explicit tower (you find the $r$!)
\[ K=K_0\subset K_1 \subset \ldots \subset K_r=L \]
with all $[K_{j+1}:K_j]=2$ and $c,d\in L$. Further, all elements of each
$K_i$ must be real. (In particular, you can't extend by $i$.)
(This is a challenging problem. Its helpful
to think in terms of polar coordinates and first find
$|\beta|=\sqrt{c^2+d^2}$.)
\item (This problem graded double.) Let $\eps=e^{2\pi i/5}$. Set $K=Q(\eps)$.
\ben
\item Find $[K:Q]$ and a basis for $K$ over $Q$.
\item Set $\gam=\eps+\eps^4$. Show that $[Q(\gam):Q]=2$ by
finding an explicit quadratic equation, with coefficients in
$Q$, satisfied by $\gam$. (Note: Since you have the basis this
is a linear algebra problem: finding a dependence between $1,\gam,\gam^2$.)
\item Use the quadratic formula to solve $\gam$ explicitly. Find an
explicit $d\in Z$ with $Q(\gam)=Q(\sqrt{d})$.
\item Show $[K:Q(\gam)]=2$. (This is immediate if you see it.)
\item As $\eps\in K$, find an explicit quadratic equation, with coefficients in
$Q(\gam)$, satisfied by $\eps$.
\item Use the quadratic formula to solve $\eps$ explicitly. (Some of the
terms will be square roots of non-real numbers, but lets allow that. The
object is to write $\eps$ in terms of usual field expressions and square
roots.)
\item Write $\eps=a+bi$. Find $a,b$ in terms of usual field expressions and square
roots, but not involving complex numbers.
\een
\pagebreak
\item Let $f(x)=x^3+ax^2+cx+d \in Q[x]$ be an irreducible cubic with one real root $\ah$
and two nonreal roots $\beta,\gam$.
\ben
\item Argue that $\gam = \ol{\beta}$. ({\tt Note:} Here, and often, we let $\ol{\kappa}$ denote
the complex conjugate of $\kappa$.)
\item Argue that $\gam\not\in Q(\ah)$.
\item Argue that $[Q(\ah,\gam):Q(\ah)]=2$.
\item Show that $\beta\in Q(\ah,\gam)$ and that $\ah\in Q(\beta,\gam)$.
\item Argue that $[Q(\ah,\beta,\gam):Q]=6$.
\item Argue that $\gam\not\in Q(\beta)$. (Idea: If $\gam\in Q(\beta)$ then
show that $Q(\ah,\beta,\gam)=Q(\beta)$ and get a contradiction.)
\een
{\tt Remark:} By the last part, $Q(\beta)$ is a field which is {\em not} closed
under complex conjugation.
\item Suppose $\ah\in C$ satisfies the equation
\[ \ah^3 + \sqrt{2}\ah^2 - (7^{1/5}-8)\ah = \sqrt{8+9\sqrt{11}} \]
Use the Tower Theorem to bound $[Q(\ah):Q]$.
\een
\beq
Dear Sir,
\\ I beg to introduce myself to you as a clerk in the Accounts Department
of the Port Trust Office at Madras on a salary of only \pounds 20 per
annum. I am now about 25 years of age. I have no University education but
I have undergone the ordinary school course. After leaving school I have
been employing the spare time at my disposal to work at Mathematics\ldots
I am striking out a new path for myself. I have made a special investigation
of divergent series in general and the results I get are termed
by the local mathematicians as ``startling" \\ -- Ramanujan
\enq
\end{document}