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\begin{document}

\textbf{Math 6320, Assignment 4  \hfill Due: March 23rd}

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\begin{enumerate}

\item 
Determine the minimal polynomial of $\cos 2\pi/7$ over $\bbQ$.

%\textbf{Solution:} 

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\item 
If $n$ is an odd positive integer, prove that $\bbQ(\cos(2\pi/n))=\bbQ(\cos(\pi/n))$.

%\textbf{Solution:} 

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\item 
Take a regular $n$-sided polygon inscribed in a circle of radius $1$. Label the vertices $P_1,\dots,P_n$, and let $\lambda_k$ be the length of the line joining $P_k$ and $P_n$ for $1\le k\le n-1$. Prove that
\[
\lambda_1\cdots\lambda_{n-1}=n\,.
\]


%\textbf{Solution:} 

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\item

Find a real number $\alpha$ such that the extension $\bbQ\subset\bbQ(\alpha)$ is Galois, with Galois group $\bbZ/6$.


%\textbf{Solution:}

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\item
Let $\alpha$ be the positive real number $5^{1/4}$. Prove that each of the extensions $\bbQ\subset\bbQ(i\alpha^2)$ and $\bbQ(i\alpha^2)\subset\bbQ(\alpha + i\alpha)$ is normal. Is $\bbQ\subset\bbQ(\alpha + i\alpha)$ normal?

%\textbf{Solution:} 

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\item
Let $E=\bbQ(e^{2\pi i/8})$. Determine all subgroups of $\gal(E/\bbQ)$, and the corresponding intermediate fields.

%\textbf{Solution:} 

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\item 
Let $\ul s:=s_{1},s_{2},s_{3}$ be the elementary symmetric polynomials in $\ul x:= x_{1},x_{2},x_{3}$. Prove that $\bbQ(\ul x)$ is not a radical extension of $\bbQ(\ul s)$, and that it is a radical extension of $\bbQ(i,\ul s)$.

%\textbf{Solution:} 


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\item
Prove that $x^{7} - 10x^{5} + 15x + 5$ is solvable by radicals over $\bbQ$.

%\textbf{Solution:} 

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\item
Let $f(x)$ be an irreducible polynomial in $\bbQ[x]$ of prime degree $p\ge 5$, and $E$ the splitting field of $f(x)$.
Prove that $f(x)$ is solvable by radicals if and only if $E=\bbQ(\alpha,\beta)$ for any pair of distinct roots $\alpha,\beta$ of $f(x)$.

%\textbf{Solution:} 

\item
Deduce from the preceding exercise that if $f(x)$ as above has exactly two real roots, then it is not solvable by radicals.
(This was proved by Galois).
%\textbf{Solution:} 

\end{enumerate}

\end{document}

