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\begin{center} {\Large\bf Honors Algebra II \\ Assignment 11} \\ Due Friday, April 22 in recitation \end{center}
\beq
It's my party, and I'll cry if I want to
\\ Cry if I want to, cry if I want to
\\ You would cry too if it happened to you
\\ -- It's My Party, Leslie Gore\\ https://www.youtube.com/watch?v=mCPqaG8sVDE
\enq
For the first two problems let $f(x)\in Q[x]$ be an irreducible
quartic with roots $\ah,\beta,\gam,\del$ and set $K=Q(\ah,\beta,\gam,\del)$,
its splitting field. We will {\em assume} $\Gam[K:Q]$ is isomorphic to
$S_4$, the full symmetric group on $4$ elements. We will represent
$\sig\in \Gam[K:Q]$ as permutations on $\ah,\beta,\gam,\del$.
For problem \ref{one} below use
the following result from Group Theory with $n=4$: The {\em only}
subgroup of $S_n$ with index two is the alternating group $A_n$.
\ben
\item
Set $\kappa = \ah\beta+\gam\del$.
\ben
\item\label{a}
Find a group $H\subset S_4$ of eight permutations
so that $\sig(\kappa)=\kappa$ for all $\sig\in H$.
\item Assume $\sig(\ah)=\ah$, $\sig(\beta)=\gam$, $\sig(\gam)=\beta$,
$\sig(\del)=\del$. Show that $\sig(\kappa)\neq \kappa$
\item Show that $Q(\kappa)^*=H$ where $H$ is the group found in (\ref{a}).
\item Deduce that $[Q(\kappa):Q]= 3$.
\een
\item Set $L=Q(\ah)$. Set $H=L^{*}$.
\ben
\item Give six $\sig\in H$.
\item Using the counting relations of the Galois Correspondence
Theorem show that $H$ consists of precisely the six $\sig$ you
described in the first part.
\item\label{one} Show that there is no group $H^+$ with $H\subset H^+\subset
S_4$ that has precisely $12$ elements.
\item Now using the Galois Correspondence Theorem show that
there is no field $M$, $Q\subset M \subset L$, with $[M:Q]=2$.
\een
\item Find $\Phi_{15}(x)$, the fifteenth cyclotomic polynomial,
explicitly. (Some grunt work here.)
\item Let $p=2k+1$ be an odd prime.
Set $\eps=e^{2\pi i/p}$ and set $K=Q(\eps)$.
Let $\tau$ denote complex conjugation and set $H=\{e,\tau\}$.
Let $L$ denote the set of real numbers in $K$.
\ben
\item Show that $L=H^{\dag}$.
\item Find $[L:Q]$.
\item Set $\gam=\eps+\eps^{-1}$. Show $Q(\gam)^*=H$.
\item Write $\gam$ in terms of trig functions.
\item Find the degree of minimal polynomial $f(x)\in Q[x]$ with
$\cos(2\pi/p)$ as a root.
\een
\een
\beq
She had been born with a map of time in her mind. She
pictured other abstractions as well, numbers and the
letters of the alphabet, both in English and in Bengali.
Numbers and letters were like links on a chain. Months
were arrayed as if along an orbit in space.
\\ Jhumpa Lahiri, The Lowland
\enq
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