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\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#3}
\date{21 January 2015; due Wednesday, 28 January 2015}

\maketitle

\begin{problab}{3.1 (M2-5)}
Let $f \in F[X]$ where $F$ is a field of characteristic zero.  Let $d=\gcd(f,f')$.  Show that $g=f/d$ has the same roots as $f$ and that all roots of $g$ are simple.  
\end{problab}

\begin{problab}{3.2 (M2-6)}
Let $f \in F[X]$ be irreducible where $F$ has characteristic $p$.  Show that $f(X)=g(X^{p^e})$ for some $e \geq 0$ and $g(X)$ is irreducible and separable.  Deduce that every root of $f(X)$ has the same multiplicity $p^e$ in any splitting field.  
\end{problab}

\begin{problab}{3.3}
Let $F \subseteq K \subseteq L$ be a tower of fields.  Suppose that $\alpha \in L$ is separable over $F$.  Show that $\alpha$ is separable over $K$.  Conclude (or argue otherwise) that if $f \in F[X]$ is separable then any splitting field of $f$ is separable over $F$.  
\end{problab}

\begin{problab}{3.4}
Let $f=X^5+X+1 \in \Z[X]$.  For a prime $p$, let $f_p \in \F_p[X]$ denote the reduction of $f$ modulo $p$.  Find all primes $p$ for which $f_p$ is not separable, and for such primes compute $\gcd(f_p,f_p')$.
\end{problab}

\end{document}