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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 10} \\ Due Friday, April 18 in recitation \end{center}
\beq
Gary's perfection hurt Burton in a physical way. He felt as he did when he watched
a theorem unfold seamlessly, the sheer elegance of it almost painful to witness
because its presence in the world threw into high relief the incomprehensible
mess of his life.
\\ from {\em Pond} by Marisa Silver
\enq
\ben
\item\label{one} Let $K:Q$ be a normal extension, set $G=\Gam[K:Q]$. Let
$H$ be a subgroup of $G$. Let $a=|H|$, $s=|G|$.
Let $\ah\in K$. Set $\gam = \sum_{\sig\in H} \sig(\ah)$.
\ben
\item Show that $\tau(\gam)=\gam$ for all $\tau\in H$.
\item Show that $H\subseteq Q(\gam)^*$.
\item \label{two} Deduce (using the Galois Correspondence Theorem) an upper bound on $[Q(\gam):Q]$.
\een
\item Let $K:Q$ be a normal extension and assume that $K$ is {\em not} a subset of
the reals.
\ben
\item Show that $K$ is closed under complex conjugation. That is, let $\ah=a+bi\in K$
with $a,b$ real. Show that $a-bi\in K$. (Warning: This is not true for any field
extension, you must use that $K:Q$ is normal.
\item Let $a+bi\in K$ with $a,b$ real. Show that $a\in K$ and $bi\in K$.
\item Let $L$ be the field of real numbers $a\in K$. Prove that $[K:L]=2$.
(One approach: Use problem \ref{one} above.)
\een
\item Let $G=(Z_4\times Z_6, +)$
\ben
\item Find two subgroups $H\subset G$ with precisely $12$ elements.
\item Find a third subgroups $H\subset G$ with precisely $12$ elements.
\item (*) Prove there aren't any more $H$.
\item Let $K:Q$ be normal with $\Gamma(K:Q)$ isomorphic to $G$ above.
Show (using the Galois Correspondence Theorem) that $K$ has precisely three distinct nontrivial squareroots.
(We count $\sqrt{a}$ and $\sqrt{q^2a}$ as the same.)
\een
\pagebreak
\item Let $\ah,\beta,\gam$ be the three roots of $x^3+2x+1=0$.
\ben
\item Find $\ah+\beta+\gam$ as an explicit integer.
\item Find $\ah^2+\beta^2+\gam^2$ as an explicit integer.
\item Find $\ah^3+\beta^3+\gam^3$ as an explicit integer.
\een
\een
\beq
It is simplicity that is difficult to make.
-- B. Brecht
\enq
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