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\begin{document}

\begin{center}
{\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ 
	{\large Assignment 7}\\Due Friday, April 9 at 08:35
\end{center}


\begin{enumerate}

	\item Let $p$ be a prime and let $K$ be a splitting field of $X^p-2\in \Q[X]$,
	so $K/\Q$ is a Galois extension.  Show that $K=\Q(a,\zeta)$ for $a\in K$
	satisfying $a^p=2$ and $\zeta\in K$ a primitive $p$ th root of unity.  Describe
	generators of $G:=\Gal(K/\Q)$ in terms of their actions on $a$ and $\zeta$,
	and describe $G$ as an abstract group (in terms of generators and relations, say).
	Write out the diagrams of intermediate fields and groups, indicating clearly the various
	containments.  Also indicate which subfields of $K$ are Galois over $\Q$.
	
	\item Let $F$ be a finite field of size $\#F$, with $K/F$ a finite extension of degree $d$.
	Prove that $K/F$ is Galois and that $\Gal(K/F)$ is a cyclic group of order $d$ with generator
	the automorphism of $K$ given by
	$$\alpha\mapsto \alpha^{\#F}.$$
	(This automorphism is called the {\em arithmetic Frobenius} map of $F$).
	
	\item This exercise gives Artin's proof of the fundamental theorem of Algebra.
	 Let $F$ be a field not of characteristic 2 and assume that all odd degree polynomials
	in $F[X]$ have a root in $F$.  Let $K$ be a quadratic extension of $F$ with the property that
	every element of $K$ has a square root in $K$.
	\begin{enumerate}
		\item Prove that any finite extension of $K$ has degree a power of $2$.  (Hint: Reduce to the Galois
		case and then consider the fixed field of the 2-Sylow subgroup of the Galois group).
		
		\item Prove that $K$ has no non-trivial finite extensions which are Galois over $F$, and conclude
		that $K$ is algebraically closed. (Hint: Use the fact that a non-trivial 2-group has an index 2 normal
		subgroup).
		
		\item Let $F=\R$ and $K=\R[X]/(X^2+1)$.  Explain (using the intermediate value theorem)
		why $F$ satisfies the hypotheses above, and using explicit formulae, show that $K$ also satisfies
		the hypotheses.  Conclude that $\C:=K$ is algebraically closed (this is the Fundamental Theorem of Algebra).
	\end{enumerate}
	
	\item Determine the Galois group of the splitting field (over $\Q$) of $X^4-14X^2+9$, and write down the lattice
	of subgroups and corresponding subfields.  Which subfields are Galois over $\Q$?
	
	\item Fix a positive integer $n$ and let $K:=\Q(\zeta_n)$ for a primitive $n$ th root of unity $\zeta_n\in \C$.
	Prove that complex conjugation $\tau\in \Aut(\C)$ restricts to an automorphism of $K$ fixing $\Q$,
	and show that the corresponding element of $\Gal(K/\Q)$ corresponds to $-1$ under the isomorphism
	$\Gal(K/\Q)\simeq (\Z/n\Z)^{\times}$.  Prove that the fixed field $K^+$ of the subgroup
	generated by complex conjugation is equal to the intersection $K\cap \R$ taken inside $\C$.
	We call $K^+$ the {\em maximal real subfield} of $K$.
	
	\item This problem works out a formula for $\cos(2\pi/17)$ in terms of square-root extractions.
	Let $\zeta:=e^{2\pi i/17}$; it is a primitive $17$ th root of unity.  Let 
	$\alpha:=\zeta+\zeta^{-1}=2\cos(2\pi/17)$.  Let $\sigma\in \Gal(\Q(\zeta)/\Q)$ be the element
	determined by
	$$\sigma \zeta = \zeta^3.$$
	
	
	\begin{enumerate}
		\item Show that $\sigma$ generates $\Gal(\Q(\zeta)/\Q)$
		\item Define the {\em periods} of $\alpha$ to be
		\begin{align*}
		& \eta_1:= \alpha + \sigma^2\alpha + \sigma^4\alpha + \sigma^8\alpha && & \eta_1':=\sigma\eta_1\\
		& \eta_2 := \alpha + \sigma^4\alpha  && & \eta_2':=\sigma^2 \eta_2 \\
		& \eta_3 := \sigma \eta_2  && & \eta_3':=\sigma\eta_2' 
		\end{align*}
		Prove that $\eta_1,\eta_1'$ are the roots of $X^2+X-4$, that $\eta_2,\eta_2'$ are the roots of
		$X^2-\eta_1X-1$, that $\eta_3,\eta_3'$ are the roots of $X^2-\eta_1'-1$
		and that $\alpha$ and $\sigma^4\alpha$ are the roots of $X^2-\eta_2X+\eta_3$.
		\item Conclude that $\cos(2\pi/17)$ is equal to
		\begin{equation*}
			\frac{1}{16}\left(-1 +\sqrt{17}+\sqrt{2(17-\sqrt{17})}
			+2\sqrt{17+3\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{2(17+\sqrt{17})}}\right)
		\end{equation*}
	\end{enumerate}
	
	\item Let $F$ be a field and $f\in F[X]$ a monic separable polynomial of degree $n$. Fix a splitting field
	$K$ of $f$ and write $G:=\Gal(K/F)$. 
	\begin{enumerate}
		\item Prove that $G$ is a subgroup of $S_n$, the symmetric group on $n$ letters.
		If $f$ is irreducible, prove that $G$ is a {\em transitive} subgroup of $S_n$
		with $\#G$ divisible by $n$.
		(Viewing $S_n$ as the permutations of an $n$-element set $T$, a transitive subgroup $G$
		is one which acts transitively on these $n$ elements, i.e. for any $x,y\in T$
		there exists $g\in G$ such that $gx=y$). 
		
		\item Prove that if $n$ is prime and $f$ is irreducible, then $G$ contains 
		an $n$-cycle.  (Hint: use Sylow's theorem.) \label{ncycle}	
		
		\item Suppose that $f$ is irreducible of degree $5$ and has exactly 3 real roots.
		Prove that $G$ is isomorphic to $S_5$.  (Hint: View $K$ as a subfield of $\C$
		and consider complex conjugation acting on $K$.  Now use
		(\ref{ncycle}) and some group theory.)
	\end{enumerate}
	
	\item Keep the notation of the previous problem.
	\begin{enumerate}
		\item Let $r_1,\ldots,r_n$ be the $n$ distinct roots of $f$ in $K$,
		and define the {\em discriminant of $f$} to be
		$$\Delta(f):= \prod_{i,j} (r_i-r_j),$$
		where the product runs over all pairs $(i,j)\in \Z^2$ with $1\le i,j\le n$.
		Prove that $\Delta(f)\in F$.
		
		\item Prove that $G$ is a subgroup of $A_n$ (the alternating group) if and only if
		$\Delta(f)$ is a square in $F$.  Hint: use the formula for $\Delta(f)$ above and the
		definition of $A_n$ as the group of even permutations.
	\end{enumerate}
	
	
\end{enumerate}

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