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\begin{document}

\textbf{Math 6320, Assignment 3  \hfill Due: Weekend of March 4th}

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\begin{enumerate}

\item For each of the following polynomials, determine the number of distinct roots in $\bbF_{49}$.
\[
x^{48}-1,\qquad x^{49}-1,\qquad x^{54}-1.
\]

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\item
Let $k$ be a field of characteristic $p>0$. Set $E=k(x,y)$ and $F=k(x^p,y^p)$.
\begin{enumerate}
\item Prove that $[E:F]=p^2$.
\item Prove that $E\neq F(\alpha)$ for any $\alpha\in E$.
\end{enumerate}

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\item 
Prove that the extension $\bbQ\subset\bbQ(\sqrt{2},\sqrt{3})$ is normal (also called Galois), and compute its Galois group.

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\item
Determine the Galois group of $x^4-2\in\bbQ[x]$. 

%\textbf{Solution:}

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\item
Determine the Galois group of $x^4+4x^2+2\in\bbQ[x]$.

%\textbf{Solution:} 

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\item
Compute the cyclotomic polynomials $\Phi_{30}(x)$ and $\Phi_{81}(x)$. Use the formula given by M\"obius inversion.

%\textbf{Solution:} 

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\item This exercise and the next give a quick method for computing  cyclotomic polynomials.
\begin{enumerate}
\item 
Prove that $\Phi_{n}(x) = x^{\vf(n)}\Phi_{n}(1/x)$, so that the coefficients of $\Phi_{n}(x)$ are palindromic. 
\item
Prove that $\Phi_{n}(x)$ is determined by its value in the residue ring $\bbZ[x]$ modulo $(x^{\lceil \vf(n)/2\rceil+1})$.
\end{enumerate}

%\textbf{Solution:} 


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\item
Prove the following identities.
\begin{enumerate}
\item 
$\Phi_{n}(x) = \Phi_{m}(x^{n/m})$ when $m$ is the product of the distinct prime factors dividing $n$.
\item
$\Phi_{pn}(x) = \Phi_{n}(x^{p})/\Phi_{n}(x)$ when $p$ is a prime not dividing $n$.
\item
$\Phi_{2n}(x) = \Phi_{n}(-x)$ when $n$ is an odd integer $\ge 3$.
\end{enumerate}

%\textbf{Solution:} 

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\item
Use the preceding exercises to compute $\Phi_{n}(x)$ for $n=30, 81, 105$; compare with your answer for (6).
\end{enumerate}

\end{document}

