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\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
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\newcommand{\ZZ}{\mathbb{Z}}
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\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
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\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{10}  % the homework number
\newcommand\DUEDATE{4/24/2020} % the due date
\newcommand\COURSE{MA 362}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
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%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


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\begin{problem}[1]
Let $\alpha = \sqrt{2 + \sqrt{2}}$.
\begin{enumerate}
\item Compute the degree $[\QQ(\alpha) :\QQ]$.
\item Prove that $\QQ(\alpha) \supseteq \QQ$ is a Galois extension.
\item Prove that the Galois group $\Gal(\QQ(\alpha) /\QQ)$ is a cyclic group.
\item Use the Fundamental Theorem of Galois Theory to draw the lattice of intermediate fields between $\QQ$ and $\QQ(\alpha)$.
\end{enumerate}
\end{problem}

%\begin{proof}
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\newpage


\begin{problem}[2]
On HW 9, you proved that the splitting field for $x^4 + 1$ over $\QQ$ is $\QQ(\zeta)$, where $\zeta^4 = -1$. You also showed that $[\QQ(\zeta) : \QQ] = 4$.
\begin{enumerate}
\item Prove that $\Gal(\QQ(\zeta) / \QQ)$ is isomorphic to the Klein $4$-group.
\item Use the Fundamental Theorem of Galois Theory to draw the lattice of intermediate fields between $\QQ$ and $\QQ(\zeta)$.
\end{enumerate}
\end{problem}

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%\end{proof}


\newpage


\begin{problem}[3]
Recall that, if $p$ is a prime number, the $p$th \textbf{cyclotomic polynomial} is
\[ \Phi_p(x) = \frac{x^p - 1}{x-1} = 1 + x + x^2 + \cdots + x^{p-1}. \]
We now define a cyclotomic polynomial $\Phi_n(x) \in \QQ[x]$ for every positive integer $n$.

Set $\Phi_1(x) = x-1$. If $n > 1$, then we inductively define $\Phi_n(x)$ so that
\[ x^n - 1 = \prod_{d | n} \Phi_d(x), \]
where the product is over all divisors $d$ of $n$. That is, we are assuming (by induction) that the cyclotomic polynomials $\Phi_d(x)$ for all \emph{proper} divisors $d<n$ have already been defined, and so the only unknown in the above equation is $\Phi_n(x)$.

\begin{enumerate}
\item Check that this inductive definition is consistent with the original definition of $\Phi_p(x)$ for $p$ prime.
\item Use this definition to compute $\Phi_8(x)$, $\Phi_{10}(x)$, and $\Phi_{12}(x)$.
\item Show that
\[ n = \sum_{d | n} \phi(d), \]
where $\phi$ denotes the Euler totient function. [\textsc{hint:} Count the number of integers $k \in \{1, \ldots, n\}$ such that $\gcd(k,n) = \frac{n}{d}$.]

\item Prove that $\Phi_n(x)$ has degree $\phi(n)$.
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[4]
Let $\Phi_n(x) \in \QQ[x]$ be the $n$th cyclotomic polynomial, as defined in the previous problem.

Recall that $\zeta$ is an \textbf{$n$th root of unity} if $\zeta^n = 1$, and $\zeta$ is a \textbf{primitive $n$th root of unity} if additionally $\zeta^i \neq 1$ for $1 \leq i < n$.

\begin{enumerate}
\item Show that $\zeta$ is an $n$th root of unity if and only if $\zeta$ is a root of $x^n - 1$, and $\zeta$ is a primitive $n$th root of unity if and only if $\zeta$ is a root of $\Phi_n(x)$.
\item Show that, for any odd positive integer $n$, $\Phi_{2n}(x) = \Phi_n(-x)$. [\textsc{hint:} If $\zeta$ is a primitive $n$th root of $1$, show that $-\zeta$ is also a root of unity. What is its order?]
\item Let $\zeta$ be a primitive $n$th root of unity. Prove that $\QQ(\zeta)$ is the splitting field of $x^n - 1$. (The field $\QQ(\zeta)$ is called the \textbf{$n$th cyclotomic field}.)

\end{enumerate}
\end{problem}

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\newpage


\begin{problem}[5]
Let $n$ be a positive integer, and let $U_n = (\ZZ/n\ZZ)^{\times}$ be the group of units in the ring $\ZZ/n\ZZ$. That is,
\[ U_n = \{ \overline{k} \in \ZZ/n\ZZ \mid \gcd(k,n) = 1 \}. \]
(You considered this group on HW 11 in MA 361.)
\\ \\
Let $\zeta$ be a primitive $n$th root of unity. 
\begin{enumerate}
\item Use the fact that $\Phi_n(x)$ is irreducible over $\QQ$ (a result due to Gauss and Dedekind) to show that $\Gal(\QQ(\zeta) / \QQ)$ is a group of order $\phi(n)$.
\item Prove that $\Gal(\QQ(\zeta) / \QQ) \cong U_n$.

[\textsc{hint:} Any automorphism of $\QQ(\zeta)$ is determined by where it maps $\zeta$. Show that there is an automorphism $\sigma_k \colon \zeta \mapsto \zeta^k$ if and only if $\gcd(k,n) = 1$.]
\end{enumerate}


\end{problem}

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