\documentclass[12pt]{amsart}

\title{Final for Field Theory}

\author{James Cummings}


\begin{document}

\maketitle


   You may consult any books or papers you wish. Please make a note of any works consulted.
   You may not collaborate but may consult with me if you have any questions.
   This final is due by email before 11:59pm on May 10. 


\begin{enumerate}


\item (20 points total)  Let $F$ be a subfield of $\Bbb C$ with $[F : {\Bbb Q}]$ finite. Prove that $F$ contains
    only finitely many roots of unity (recall that a root of unity is a complex number
    $\zeta$ such that $\zeta^n = 1$ for some $n > 0$).


\item (60 points total) Let $K$ be the subfield of $\mathbb C$ generated by
  the complex roots of $x^5 + 2$. 
  
\begin{enumerate}

\item  (5 points)  Find $[K : {\Bbb Q}]$.

\item  (10 points) Describe the group $G = Aut(K/{\Bbb Q})$.

\item  (10 points) Give an explicit isomorphism between $G$ and a subgroup $H$ of $S_5$. 
 
\item   (25 points) Find all the subgroups of $G$ and all the fields intermediate between ${\Bbb Q}$ and $K$.
     Which ones are Galois extensions of ${\Bbb Q}$?  Identify $K \cap {\Bbb R}$. 

\item  (10 points) Find all the roots of unity in $K$. 

\end{enumerate} 


\item(25 points total) 

\begin{enumerate} 

\item   (15 points)  Let $F$ be a finite degree extension of $E$, and assume $F$ is a  Galois extension
     of $E$.  Say that $F_1$ is an ``abelian extension of $E$'' if $F_1$ is a subfield
     of $F$ containing $E$, $F_1$ is a Galois extension of $E$ and $Aut(F_1/E)$ is abelian.
     Show that there is a unique abelian extension $F_{max}$ of $E$ which contains all the
     abelian extensions.  Hint: use the fundamental theorem to see which group
     this field must correspond to!  

\item  (10 points)  Let $E = {\Bbb Q}$ and $K$ the field considered in the
     preceding question.  Find $K_{max}$. 

\end{enumerate}


\item  (25 points total)   Prove that for any finite group $G$ there is a pair of fields $E$ and $F$ such that
    $F$ is a Galois extension of $E$ and  $Aut(F/E)$ is isomorphic to $G$. 
   Hint: any finite group is isomorphic to a subgroup of $S_n$. 


\item (10 points total)   Find the minimal polynomial of $1 + \sqrt{2} + \sqrt[3]{3}$ over $\Bbb Q$. 


\item (40 points total)  Let $\alpha$ be a real irrational number which is algebraic over $\Bbb Q$,
    let $f$ be the minimal polynomial of $\alpha$ over $\Bbb Q$ and let
    $n$ be the degree of $f$.
\begin{enumerate}
\item  (10 points)  Prove that $f'(\alpha) \neq 0$, and that $f(r) \neq 0$ for all
   rational $r$.
\item  (10 points) Prove that there is a constant $D>0$ such that
   for every $p, q \in {\Bbb Z}$ with $q > 0$,
   $\vert f(p/q) \vert \ge D/q^n$.
  Hint: you may find it easier to think about multiplying $f$ by some
    integer to get a polynomial in ${\Bbb Z}[x]$.
\item (10 points)  Prove there is a constant $C > 0$ such that 
   for every $p, q \in {\Bbb Z}$ with $q > 0$,
   $\vert \alpha - p/q \vert \ge C/q^n$.
  Hint:  the mean value theorem.
\item (10 points) Prove that $\sum_{n=0}^\infty 10^{-n!}$ is transcendental over
   $\Bbb Q$.
\end{enumerate}


\item (50 points total)   Let $F_0 = {\mathbb Q}$.  For each $n > 0$ let $\alpha_n = \sqrt[2^n]{2}$, and let $F_n = {\mathbb
  Q}(\alpha_n)$.

\begin{enumerate}

\item  (10 points) Prove that $F_n \subseteq F_{n+1}$ and find $[F_{n+1}: F_n]$.  


\item   (20 points) Let $F_\infty = \bigcup_n F_n$. Find a basis for $F_\infty$ as a 
   vector space over $F_0$, and use it to show that $F_\infty$ has countably
   infinite dimension as an $F_0$-vector space.

\item   (20 points) Is $Aut(F_\infty/F_0)$ finite, countably infinite or
  uncountably infinite? 

\end{enumerate} 












\end{enumerate}






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