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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 9} \\ Due Friday, April 11 in recitation \end{center}
\beq
She guessed at it all, what might live, moving purposefully or
drifting aimlessly, under the deep water around her, but she
didn't think too much about any of it. It was enough to be
aware of the million permutations possible around her, and
take comfort in knowing she would not, and really could not,
know much at all.
\\ Dave Eggers, The Circle
\enq
\ben
\item Let $F\subset L \subset K$ be fields in $C$. Assume that
$K:F$ is normal {\em and} that $L:F$ is normal. Let $\tau\in \Gam[K:F]$
and $\sig\in \Gam[K:L]$. Let $l\in L$
\ben
\item Using a result shown in class argue that $\tau(l)\in L$.
\item Show that $(\tau\sig\tau^{-1})(l) = l$.
\item From the above show that $\Gam[K:L]$ is a {\em normal} subgroup
(get out those Algebra I notes!) of $\Gam[K:F]$. (Assume its already
been shown that it is a subgroup. You only need show the normal part.)
\item In the case $F=Q$, $L=Q(\omega)$, $K=F(\ah,\omega)$ (with
$\ah=2^{1/3}, \omega = e^{2\pi i/3}$ as in our standard example)
give the groups $\Gam[K:L]$ and $\Gam[K:F]$ explicitly in terms of
permutations of $\ah,\beta=\ah\omega, \gam=\ah\omega^2$.
\een
\item Let $F\subset K$ be fields in $C$ with $[K:F]=2$. Prove that
$K:F$ is a normal extension.
\item Let $K_1,K_2$ be normal extensions of $Q$. Let $M$ denote the
minimal field containing $K_1\cup K_2$. Prove that $M$ is a normal
extension of $Q$. [One approach: Write $K_1=Q(\ah_1,\ldots,\ah_r)$
where the $\ah_i$ are all the roots of some $p(x)\in Q[x]$ and
similarly write $K_2=Q(\beta_1,\ldots,\beta_s)$.
\item Let $\ah$ be a root of $f(x) = x^3 + x^2 - 2x - 1 \in Q[x]$.
\ben
\item Show $f(x)$ is irreducible over $Q$. {\tt Note:} You should assume this in what follows.
\item Find $[Q(\ah) : Q]$.
\item Set $\beta=-1/(\ah+1)$. Find $\beta$ in the form $a+b\ah+c\ah^2$.
\item Show that $f(\beta)=0$. (Bit of grunt work here!)
\item Find $\gam\in Q(\ah)$, $\gam,\beta,\ah$ distinct, with $f(\gam)=0$.
(Idea: If $f(x)=(x-\ah)(x-\beta)(x-\gam)$ then $\ah+\beta+\gam$ is determined.)
\item Deduce that $Q(\ah) : Q$ is normal.
\item
Argue that $\Gamma(Q(\ah):Q)$ has precisely three elements. Which permutations
of $(\ah,\beta,\gam)$ do the three automorphisms correspond to? What well known
group is $\Gamma(Q(\ah):Q)$ isomorphic to? (Hint: There is only one group on
three elements!)
\item Let $K$ be the fixed field of $\Gamma(Q(\ah):Q)$. Prove that $K=Q$.
\een
\item As in last week's assignment set
$\ah=2^{1/4}$, $\beta=i\ah$, $\gam=-\ah$, $\del=-i\ah$. Set
$p(x)=x^4-2$. Set $K=Q(\ah,\beta,\gam,\del)$. Set $L=Q(i)$.
Also, set $M=Q(\ah)$ and $N=Q(\sqrt{2})$.
\ben
\item Give the factorization of $p(x)$ into irreducible factors in $Q[x]$.
\item Give the factorization of $p(x)$ into irreducible factors in $K[x]$.
\item Give the factorization of $p(x)$ into irreducible factors in $M[x]$.
\item Give the factorization of $p(x)$ into irreducible factors in $N[x]$.
\een
\een
\beq
Tell me, what do you plan to do with your one wild and precious life?
\\ Mary Oliver, The Summer Day
\enq
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