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\parbox{2.25in}{\bf Math 4010/5530}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Apr. $\mathbf{6}^\mathrm{\bf th}$, 2016}

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%\noindent {\bf Name:} \  \underline{\qquad Will Conley\qquad}

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\begin{description}
\item[Page 63 \#79] Let $\mathbb{F}$ be a field, $f(x) \in \mathbb{F}[x]$, and $\mathbb{E}$ be a splitting field for $f(x)$ over $\mathbb{F}$. In addition, let $G=\mathrm{Gal}(\mathbb{E}/\mathbb{F})$.
   \begin{enumerate}[(a)]
   \item Suppose that $f(x)$ is irreducible. Show that $G$ acts {\bf transitively} on the roots of $f(x)$ in $\mathbb{E}$.  This means that given any two roots $\alpha, \beta \in \mathbb{E}$ of $f(x)$, there exists some $\sigma \in G$ such that $\sigma(\alpha)=\beta$. {\it Hint:} Consider the lemma and theorem on pages 55 \& 56.
   \item Suppose that $f(x)$ has no repeated roots and that $G$ acts transitively on the roots of $f(x)$ in $\mathbb{E}$. Show that $f(x)$ is irreducible. {\it Hint:} Suppose that $f(x)$ factors. $G$ must send roots of a factor to other roots of that factor. Transitivity of the action will force distinct factors to share a root (why?). This is a problem (why?).
   \end{enumerate}
   
   This exercise then proves that for a separable polynomial $f(x)$, $G$ acts transitively on the roots of $f(x)$ iff $f(x)$ is irreducible.

\item[Page 63 \#80] Let $f(x)=x^4-10x^2+1 \in \mathbb{Q}[x]$. Find the Galois group of $f(x)$ (over $\mathbb{Q}$).\\ {\it Note:} The roots of $f(x)$ are $\pm\sqrt{2}\pm\sqrt{3}$. Rotman suggests looking at Exercise 67 (page 43) and Example 20 (page 54). My suggestive calculation: $(\sqrt{2}+\sqrt{3})^2=5+2\sqrt{6}$. $\sqrt{6}(\sqrt{2}+\sqrt{3})=3\sqrt{2}+2\sqrt{3}$. $3(\sqrt{2}+\sqrt{3})-(3\sqrt{2}+2\sqrt{3})=\sqrt{3}$.
   
\item[Roots of Unity] Just remember $\sqrt{1}=1$. 
    \begin{enumerate}[(a)]
    \item Find a primitive $n^{\mbox{th}}$-root of unity when $n=1,2,3,4,5,$ and $6$. Your final formulas should not involve sines and cosines. For example: $e^{2\pi i/3} = \cos(2\pi/3)+i\sin(2\pi/3)$ is a primitive third root of unity, but an unacceptable (final) answer. You may find Wolfram Alpha helpful as you seek out what numbers like $\cos(2\pi/5)$ are (unless you have more special triangles memorized than I do).
    
    %\begin{tabular}{c||c|c|c|c|c|c|}
    %$n=$  & 1 & 2 & 3 & 4 & 5 & 6 {\large\strut} \\ \hline
    %Root & & & & & & {\Huge\strut} \\ \hline
    %\end{tabular}
    
    \item Find the Galois  groups of $x^n-1$ over $\mathbb{Q}$ when $n=1,2,3,4,5,6$ and $7$. 
    
    %\begin{itemize}
    %\item $\mathrm{Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}) \cong$ ??? when $\zeta = e^{2\pi i/1}=1$ because ???
    %\item $\mathrm{Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}) \cong$ ??? when $\zeta = e^{2\pi i/2}=-1$ because ???
    %\end{itemize}
   
    \end{enumerate}
\end{description}

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