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\title{Algebra Homework Set 9}
\author{James Cummings (jcumming@andrew.cmu.edu)}


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   Due by class time on Wednesday 30 November. Homework must be typeset and submitted by email as a PDF file. 
   Note: Some of these use the ``Fundamental theorem of Galois theory'', which was stated in class but not yet proved. 
   
\begin{enumerate}

\item  $F$ is said to be an {\em abelian} extension of $E$ if $F$ is a Galois extension of $E$ and
   $Aut(F/E)$ is an abelian group.
  Prove that if $[F : E] < \infty$  and $F$ is a Galois extension of $E$, there is an intermediate
  field $F^{\rm ab}$ such that
  \begin{itemize}
  \item $F^{\rm ab}$ is an abelian  extension of $E$.
  \item For every intermediate field $F'$ which is an abelian extension of $E$, $F' \subseteq F^{\rm ab}$.
  \end{itemize} 


\item Prove that if $[F : E] < \infty$ and $F$ is a Galois extension of $E$, then 
  for every $a \in F$ we have $m_a^E = \prod_{i =1}^n (x - a_i)$, where $a_1, \ldots a_n$
  enumerates the orbit of $a$ under the action of $Aut(F/E)$.

  
\item (Eisenstein criterion) Let $p$ be prime and let $f = x^n + \sum_{i < n} a_i x^i \in {\mathbb Z}[x]$
  where $n > 1$, $p \vert a_i$ for $0 \le i < n$, $p^2 \not\vert a_0$. Prove that $f$ is irreducible in ${\mathbb Q}[x]$.
  Hint: By Gauss ETS it is irreducible in ${\mathbb Z}[x]$. If $f = g h$ then reduce mod $p$ to get
  an equation $x^n = {\bar g} {\bar h}$ in the UFD  ${\mathbb Z}/p {\mathbb Z}[x]$.  

\item By the last question $f = x^3 - 2$ is irreducible in ${\mathbb Q}[x]$. Let
  $\alpha = 2^{1/3}$, $\zeta = \exp(2 \pi i/3)$ and let $F = {\mathbb Q}(\alpha, \zeta)$.

  \begin{enumerate}
  \item Show that $F$ is a splitting field for $f$ over $\mathbb Q$.
  \item Compute $[F: E]$.
  \item Give a basis for $F$ as a VS over $E$. 
  \item Describe the structure of $Aut(F/E)$. 
  \item Find all the intermediate fields and identify the ones which are Galois extensions of $E$.
  \item Identify the field $F^{\rm ab}$ from Q1.
  \end{enumerate}
  
\item As usual let ${\mathbb Q}(x)$ denote the field of fractions of the ID ${\mathbb Q}[x]$.
  Let $\sigma$ be the automorphism of ${\mathbb Q}(x)$ which maps $x$ to $1/x$, let $\tau$
  be the the automorphism of ${\mathbb Q}(x)$ which maps $x$ to $1 -x$, and let $H$ be the
  group generated by $\sigma$ and $\tau$. Finally let $F = {\mathbb Q}(x)$ and  $E = Fix(H)$.

  \begin{enumerate}
  \item Describe the structure of $H$.
  \item Compute $[F:E]$.
  \item Prove that $F = E(x)$ and find the minimal polynomial of $x$ over $E$. 
  \item Find some $a \in F$ such that $E = {\mathbb Q}(a)$.
  \end{enumerate}

  
\item Prove that:
  \begin{enumerate}
  \item ${\mathbb R}$ has no extensions of odd degree greater than one. Hint: Intermediate value theorem.
  \item Up to isomorphism, ${\mathbb R}$ has only one extension of degree $2$.  
  \end{enumerate}

\item Let $[F: E] < \infty$ with $F$ a separable extension of $E$. Prove that
  \begin{enumerate}
  \item There is an extension $F'$ of $F$ such that $[F' : F] < \infty$ and
    $F'$ is a Galois extension of $E$.
  \item There are finitely many fields intermediate between $E$ and $F$.
  \end{enumerate}
  

  
  
\end{enumerate}

  
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