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\begin{center} {\Large\bf Algebra V63.0349 \\ Assignment 9} \\ Due Friday, April 17 in recitation \end{center}
\beq
I doubt sometimes whether a quiet and unagitated life would have
suited me -- yet I sometimes long for it. \\ -- Byron
\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 $[K:L]=2$.
\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
\item Recall that $\ah$ is square root constructible if there
exists a sequence $u_1,\ldots,u_s=\ah$ such that
each $u_i$ is either in $Q$ or the sum or product or
additive inverse or multiplicative inverse or -- critically --
a squareroot of previous element(s). Equivalently (we
showed this in class, you can assume it) $\ah$ is square root
constructible if there
exists a tower of fields $Q=F_0\subset F_1\subset\cdots F_r=F$
with all $[F_{i+1}:F_i]=2$ and $\ah\in F$.
Our object here is to
give necessary {\em and} sufficient conditions on $\ah$ being
square root constructible.
\ben
\item Let $\ah$ be square root constructible by the sequence
$u_1,\ldots,u_s=\ah$. Set $K=Q(u_1,\ldots,u_s)$. Let $\sig: K\ra K_1$
be an isomorphism over $Q$. Prove $\sig(\ah)$ is
square root constructible by the sequence
$\sig(u_1),\ldots,\sig(u_s)=\sig(\ah)$.
\item Let $\ah$ have minimal polynomial $p(x)\in Q[x]$ and let
$\beta$ be a root of $p(x)$. Let $\ah$ be square root constructible.
Prove that $\beta$ is square root constructible.
\item Let $\ah$ have minimal polynomial $p(x)\in Q[x]$, with roots
$\ah=\ah_1,\ldots,\ah_s$. Let $K=Q[\ah_1,\ldots,\ah_s]$ be the
splitting field of $p(x)$ over $Q$. Prove that if $\ah$ is
square root constructible then every $\gam\in K$ is square root
constructible. Prove further that $[K:Q]$ must be a power of two.
\item Let $\ah$ have minimal polynomial $p(x)\in Q[x]$, with roots
$\ah=\ah_1,\ldots,\ah_s$. Let $K=Q[\ah_1,\ldots,\ah_s]$ be the
splitting field of $p(x)$ over $Q$. Now {\em assume} $[K:Q]=2^w$
for some integer $w$. Prove that all $\gam\in K$ (so that includes
$\ah$ itself!) are square root constructible.
(Hint: Use the Galois Correspondence Theorem
and the Sylow Theorem 2.12.1 on existence of subgroups.)
\een
\een
\beq
It is simplicity that is difficult to make.
-- B. Brecht
\enq
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