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\title{Math 721\\Spring 2011\\Exam 2}
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{\it Due Monday April 18, 2011. As is usual on exams, I am the only biological resource that you should use.}
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\noindent 1. Consider the polynomial $f(x)=x^5-3\in \mathbb{Q}[x]$. 
\begin{itemize}
\item[a)] (5 pt) Let $F$ be  the field obtained by adjoining one of the complex roots of the above polynomial to $\mathbb{Q}$. Find $[F:\mathbb{Q}]$ and show that this extension is not Galois.
\item[b)] (5 pt) Show that $x^4+x^3+x^2+x+1$ is irreducible over $F$.
\item[c)] (5 pt) If $\overline{F}$ is the field obtained by adjoining all of the roots of $f(x)$ to $\mathbb{Q}$, find the Galois group $\text{Gal}(\overline{F}/\mathbb{Q})$. (Hint: this group must be a transitive subgroup of $S_5$.) 
\end{itemize}

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\noindent 2. Consider the field, $F$, obtained by adjoining all roots of the polynomial $f(x)=x^6-4x^3+1$ to the rational numbers $\mathbb{Q}$.
\begin{itemize}
\item[a)] (5 pt) Show that complex conjugation is a nontrivial automorphism of $F$.
\item[b)] (5 pt) If $\alpha$ is a real root of this polynomial, show that the map induced by $\alpha\mapsto\alpha^{-1}$ gives rise to an automorphism of $\mathbb{Q}(\alpha)$.
\item[c)] (5 pt) Show that $[F:\mathbb{Q}]=12$.
\item[d)] (5 pt) Find the Galois group of $F$ over $\mathbb{Q}$.
\end{itemize}

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\noindent 3. (5 pt) Consider the field extension $\mathbb{Q}\subseteq\mathbb{Q}(x)$. Show that $\mathbb{Q}(x^2)$ is a closed intermediate extension, but $\mathbb{Q}(x^3)$ is not.

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\noindent 4. (5 pt) Show that if $K$ is a field such that $\text{char}(K)\neq 2$ and $[F:K]=2$ then $F$ is Galois over $K$.

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\noindent 5. (5 pt) If $E$ is Galois over $K$ and $F$ is Galois over $E$, is it true that $F$ is Galois over $K$? Prove the statement or give a counterexample.  
  


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