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\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#6}
\date{11 February 2015; due Wednesday, 18 February 2015}

\maketitle

\begin{problab}{6.1}
Recall that $V=\{(),(1\ 2)(3\ 4),(1\ 3)(2\ 4),(1\ 4)(2\ 3)\} \trianglelefteq S_4$ is a normal subgroup.  
\begin{enumalph}
\item Show that any transitive subgroup $G \leq S_4$ is equal to one of $S_4,A_4,V$ or is isomorphic to either $D_8$ (three conjugate subgroups) or $\Z/4\Z$ (three conjugate subgroups).  \emph{[Hint: see Figure 8 on page 110 of Dummit and Foote.]}
\item Suppose that $G \leq S_4$ is a transitive subgroup.  Prove that the indices in the following table are correct.
\[ \begin{array}{c||c|c}
G\cong & \#(G \cap V) & [G : V \cap G] \\
\hline
S_4 & 4 & 6 \\
A_4 & 4 & 3 \\
V & 4 & 1 \\
D_8 & 4 & 2 \\
\Z/4\Z & 2 & 2
\end{array} \]
\item Compute the Galois groups of the following polynomials:
\begin{center} 
$f_1(X)=X^4-X+1$, \quad $f_2(X)=X^4-X^3+X^2-X+1$ \\ 
$f_3(X)=X^4-X^3+2X^2+X+1$, \quad $f_4(X)=X^4-2X^3+2X^2+2$.
\end{center}
\item For each of the polynomials in part (c), and for each partition $\lambda$ of $4$, count the proportion of primes $p \leq 10^5$ with $p \nmid D(f)$ such that the factorization of $f_i$ modulo $p$ is given by $\lambda$.  Assuming that these proportions are rational numbers with denominator dividing $\#\Gal(f_i)$, give a conjecture for what they are (and how they relate to $G$). 
\end{enumalph}
\end{problab}

\begin{problab}{6.2 (M4-1)}
\begin{enumalph}
\item What is the splitting field of $X^m-1$ over $\F_p$?
\item Show that there is a field homomorphism $\F_{p^r} \hookrightarrow \F_{p^s}$ if and only if $r \mid s$.
\end{enumalph}
\end{problab}

\begin{problab}{6.3}
Let $p$ be prime and define
\[ a_n(p)=\#\{f \in \F_p[X]:\deg f=n,\ f \text{ monic irreducible}\}. \]
\begin{enumalph}
\item Show that $a_2(p)=(p^2-p)/2$ and $a_3(p)=(p^3-p)/3$.
\item Use the equality
\begin{equation*}
\sum_{d \mid n} da_d(p) = p^n \tag{$*$}
\end{equation*}
(which you may assume) to compute $a_n(2)$ for $n=1,\dots,5$.
\item Use $(*)$ to prove that
\[ \frac{p^n-2p^{n/2}}{n} < a_n(p) \leq \frac{p^n}{n}. \]
Conclude that the probability that a random monic polynomial of degree $n$ over $\F_p$ is irreducible is roughly $1/n$.   (This is like the ``prime number theorem'' for $\F_p[X]$.)
\end{enumalph}
\end{problab}

\begin{problab}{6.4 (M4-9)}
Let $f(X)$ be an irreducible polynomial in $\Q[X]$ with both real and nonreal roots.  Show that its Galois group is nonabelian.  Can the condition that $f$ is irreducible be dropped?
\end{problab}

\begin{problab}{6.5}
Let $\alpha=\sqrt[3]{2}$ and $\omega=(-1+\sqrt{-3})/2$.  Show that $\omega+c\alpha$ is a primitive element for $K=\Q(\alpha,\omega)$ for all $c \in \Q^\times$.
\end{problab}

\end{document}