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\begin{document}

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\centerline{\textbf{Dartmouth College}} \centerline{Mathematics 81/111
  --- Homework 7}

\begin{questions}
  \question Consider the splitting field $L$ of $x^4 - 2$ over $\Q$.
  The goal will be to characterize the Galois group and to write down
  the complete lattice of subgroups of the Galois group and the
  corresponding lattice of intermediate fields of the splitting field.
  \begin{parts}
    \part Start by writing down some elements of the Galois group and
    prove they generate the whole group.  Determine the isomorphism
    class of this group.
    \part Next draw (and label) the lattice of subgroups of the Galois
    group.
    \part Finally for each subgroup $H$, find generators for the
    corresponding fixed field so they have the form:
    $L^H = \Q(\alpha)$ or $\Q(\alpha,\beta)$.  Identify conjugate
    fields and the automorphism that relates them.
  \end{parts}


  \question Suppose that $f\in \Z[x]$ is an irreducible quartic whose
  splitting field over $\Q$ has Galois group isomorphic to the
  symmetric group $S_4$.  You may find the lattice of subgroups of
  $A_4$ given below useful in this problem.

  \begin{parts}
    \part Show that $A_4$ is the only subgroup of index 2 in $S_4$.


    \part Let $\theta$ be a root of $f$, and set $K = \Q(\theta)$.
    Prove that $K/\Q$ is an extension of degree four which contains no
    proper subfields.
 

    \part Are there any Galois extensions of degree four which contain
    no proper subfields?

  \end{parts}
 


\begin{figure}[h]
  \centering
  {\label{fig:nonsingular}\includegraphics[width=\textwidth]{A4}}

  \caption{Lattice of Subgroups of $A_4$ from Dummit and Foote}
 
\end{figure}


\question Let $L/K$ be a finite Galois extension of degree $n$, and
put $G = Gal(L/K)$.  Let $\alpha \in L$ and let $f = min_{\alpha,K}$
be its minimal polynomial, say of degree $d$.  Let
$h = \prod_{\sigma \in G} (x - \sigma(\alpha))$.  Show that
$h = f^{n/d}$.







\question Let $f \in \Q[x]$ be a polynomial of degree $n \ge 3$, and
let $K$ be the splitting field of $f$ over $\Q$. Suppose that
$\Gal(K/\Q) \cong S_n$, the symmetric group.
\begin{parts}
  \part Show that $f$ is irreducible.
  \part If $\alpha$ is a root of $f$ in $K$, show that
  $\Aut(\Q(\alpha)/\Q)$ is trivial.
  \part If $n\ge 4$ and $\alpha$ a root of $f$, show that
  $\alpha^n \not\in \Q$.
\end{parts}


\vspace{.5in}

\textbf{Some background.}  As preface to the next two problems,
consider the following situation.  Let $f \in K[x]$ be a separable
polynomial of degree $n$, $L$ its splitting field over $K$, and
$G = \Gal(L/K)$.  In the last homework set we showed that $G$ embeds
into the symmetric group via a homomorphism induced by the action of
$G$ on the $n$ distinct roots, $\alpha_1, \dots, \alpha_n$ of $f$.
That is, we showed that each element $\sigma \in G$ permutes the
$\alpha_i$ (transitively if $f$ is irreducible) and that $\sigma$ is
completely determined by this action.  This allows us to embed $G$ as
a subgroup of the symmetric group $S_n$.

Now viewing $\sigma$ as a permutation, we can write $\sigma$ in two
useful fashions: (uniquely) as the product of disjoint cycles, or as a
product of transpositions, where the only uniqueness in the later
expression is that the number of transpositions always has the same
parity.  Permutations are called \textbf{even} or \textbf{odd}
depending on the parity of this expression, and the set of even
permutations forms a normal subgroup of index 2 in $S_n$ called $A_n$,
the alternating group.

So given $\sigma \in S_n$ consider its representation as a product of
$t$ disjoint cycles (including 1-cycles) having lengths
$n_1, \dots, n_t$.  These cycle lengths form a partition of $n$, that
is $n = n_1 + \cdots + n_t$.  Also suppose that $\sigma$ can be
written as the product of $s$ transpositions.

Then the \textbf{sign} of the permutation $\sigma$,
$\sign(\sigma) = \pm1$, can be defined equivalently as

\[\sign(\sigma) = (-1)^{n-t} = (-1)^s.
\]

\question Now suppose that $f$ is a monic, separable polynomial in
$K[x]$, $L$ its splitting field, and $\{\alpha_1, \dots, \alpha_n\}$
its distinct roots. Define
\[\Delta(f) = \prod_{i<j}(\alpha_i - \alpha_j),\qquad D(f) =
  \Delta(f)^2 = \prod_{i<j}(\alpha_i - \alpha_j)^2.
\]
The quantity $D(f)$ is called the \textbf{discriminant} of $f$.
\begin{parts}
  \part Show that for an element $\sigma \in S_n$,
  $\sigma(\Delta(f)) = \sign(\sigma)\Delta(f)$.
  \part Still identifying $\Gal(L/K)$ with its image in $S_n$, show
  that $D(f) \in K$.
  \part With the same identification of $\Gal(L/K) \subseteq S_n$,
  show that $\Gal(L/K) \subseteq A_n$ if and only if
  $\Delta(f) \in K$.
\end{parts}


\textbf{More background.}  If $f \in \Z[x]$ is a monic polynomial,
then one can show $D(f) \in \Z$, and (as is clear from its definition)
is nonzero iff $f$ is separable.  As a consequence if we consider $f$
modulo $p$, it is separable iff $D(f) \not\equiv 0 \pmod p$, so if $f$
is separable, so is its reduction mod $p$ for all but the finite
number of primes $p \mid D(f)$.
    
So suppose that $p \nmid D(f)$.  Then we can write
\[ f \equiv \overline f_1 \overline f_2 \dots \overline f_t \pmod
  p, \] where the $\overline f_i$ are irreducible in $\F_p[x]$ of
degree $n_i$, so that the decomposition type of $f$ modulo $p$
($n_1, \dots, n_t$) is another partition of $n$.  Note that I have
purposely used the same notation $n_i$ as well as the number $t$ in
both here and in the earlier discussion.  This is justified by the
theorem of Frobenius.




{\bf Theorem. }(Frobenius) The density of the set of primes $p$ for
which $f$ has decomposition type $n_1, \dots, n_t$ (modulo $p$)
exists, and is equal to the number of $\sigma \in G$ ($G$ the Galois
group of $f$) with cycle type $n_1, n_2, \dots, n_t$.

So in particular, if
$f \equiv \overline f_1 \overline f_2 \dots \overline f_t \pmod p$ as
above, there is an element $\sigma \in G$ with cycle type
$n_1, \dots, n_t$.
    
\question Let $f(x) = x^4 + 30x^2 +45$.  Show that the Galois group
$G$ of $f$ over $\Q$ is cyclic of order 4.  As a small hint, it is
easy to show $f$ is irreducible over $\Z$, and hence viewing $G$ as a
subgroup of $S_4$, it must be a transitive subgroup, and therefore one
of $\Z_2\times\Z_2, \Z_4, D_8\textrm{ (the group of order 8)}, A_4$,
or $S_4$.  A starting point would be to show $|G| \le 8$.




\end{questions}

\end{document}



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