\documentclass[12pt]{exam}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{fullpage, paralist}
\usepackage{subfig}
\usepackage{graphicx}
\usepackage[all]{xy}

\CompileMatrices



\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\F}{{\mathbb F}}

\newcommand{\A}{{\mathcal A}}
\newcommand{\B}{{\mathcal B}}

\newcommand{\I}{{\mathcal I}}
\newcommand{\cC}{{\mathcal C}}

\DeclareMathOperator{\Gal}{Gal}


\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newtheorem{thm}{Theorem}[section]

\theoremstyle{remark}
\newtheorem*{nnremark}{Remark}



\begin{document}

\parskip=\medskipamount \thispagestyle{empty}

\begin{flushright}
  Your Name Goes Here
\end{flushright}
\vspace{.5in}

\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 6}

\begin{questions}

  \question Let $K$ be the splitting field over $\Q$ of
  $f(x) = (x^2 - 2)(x^2 - 3)(x^2-5)$.
  \begin{parts}
    \part Characterize the Galois group $\Gal(K/\Q)$, i.e., describe
    the elements and the isomorphism class of the group.

    \part Determine all the intermediate fields $F$, with
    $\Q \subseteq F \subseteq K$, and their degrees.

    \part Prove that $\xi = \sqrt2 + \sqrt 3 + \sqrt 5$ is a primitive
    element for the extension. Hint: Do this in a Galois way, that is,
    do not try to compute the degree of $\xi$ over $\Q$.
  \end{parts}
  
  \question Let $p$ be a prime, and $K$ the splitting field of
  $x^p - 2$ over $\Q$.
  \begin{parts}
    \part Characterize the elements of the Galois group of $K/\Q$
    ensuring of course that the maps you write down are indeed
    automorphisms.

    \part Show that $\Gal(K/\Q)$ is isomorphic to the group of
    matrices $\left(\begin{smallmatrix} u&v\\0&1
      \end{smallmatrix}\right)$,
    where $u,v \in \F_p$, $u \ne 0$.

  \end{parts}

  \question Let $\phi$ be the Euler totient function, which we can
  define by $\phi(1) = 1$ and for $n \ge 2$,
  $\phi(n) = \#(\Z/n\Z)^\times$. We know by the Chinese Remainder
  Theorem, if $m,n \ge 2$ are relatively prime integers then
  $\Z/mn\Z \cong \Z/m\Z \times \Z/m\Z$ as rings, so their unit groups
  are also isomorphic:
  $(\Z/mn\Z)^\times \cong (\Z/m\Z)^\times \times (\Z/n\Z)^\times$.
  Thus the Euler function satisfies $\phi(mn) = \phi(m)\phi(n)$ when
  $\gcd(m,n) = 1$.

  \begin{parts}
    \part Show that for any two positive integers $m,n$ we have
    $\phi(m)\phi(n) = \phi([m,n]) \phi((m,n))$ where $[m,n]$ denotes
    the lcm of $m,n$ and $(m,n)$ denotes their gcd.


    \part Show that composite of the cyclotomic fields
    $\Q(\zeta_m) \Q(\zeta_n)$ is the cyclotomic field
    $Q(\zeta_{\ell})$ where $\ell = [m,n]$.

    \part Using the above and some Galois theory, show that
    $\Q(\zeta_m) \cap \Q(\zeta_n) = \Q(\zeta_{d})$, where $d = (m,n)$.
    In case we have not quite gotten there, Theorem VI.1.12 should be
    useful.
  \end{parts}

  \question Let $L/K$ be a finite extension of fields. Let $[L:K]_s$,
  the separable degree of $L/K$, denote the number of distinct
  embeddings of $L/K$ into an algebraic closure $\overline K$.  Let
  $L_s = \{ \alpha \in L \mid \alpha \textrm{ is separable over }
  K\}$.  We are going to explore inseparability a bit more.  You could
  find most of this in V.6 of Lang, but since we did not do it in
  class, it is worth your time to work through this exercise on your
  own.

  If $L/K$ is separable, we know that $L=L_s$, so let's assume that
  $L/K$ is not separable.  Let $\sigma: L_s/K\to \overline K$ be an
  embedding.  Characterize all the extensions of $\sigma$ to an
  embedding of $L/K \to \overline K$.  Use this to show that
  $[L:K]_s = [L_s:K]$.
 
  
  \vfill \rightline{(assignment continued on next page)}
  \newpage
  \question Let $L/K$ be a finite separable extension of fields with
  $[L:K] = n$, and let $\widehat L$ be the normal (i.e., Galois)
  closure of $L$.  We have shown in class that
  $[\widehat L:K] \le n!$.  Now establish the sharper result that
  $[\widehat L:K] \mid n!$.  Hint: If we let $G = \Gal(\widehat L/K)$,
  the result would follow (why?) by showing the existence of an
  injective homomorphism $\varphi: G \to S_n$.






\end{questions}

\end{document}







%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
