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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 6} \\ Due,
Friday, Mar 7 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
\begin{center}{\bf Reminder: MIDTERM, Wednesday, March 12 in class }\end{center}
\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.
\een
\beq
Nothing is more fruitful - all mathematicians know it - than those
obscure analogies, those disturbing reflections of one theory in
another; those furtive caresses, those inexplicable discords; nothing
also gives more pleasure to the researcher. The day comes when the
illusion dissolves; the yoked theories reveal their common source
before disappearing. As the {\em Gita} teaches, one achieves
knowledge and indifference at the same time.
\\ Andr\'e Weil
\\ (Note: ``indifference" is a controversial translation of the original Sanskrit, ``detachment" is often
used instead)
\enq
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