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\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#2}
\date{14 January 2015; due Thursday, 22 January 2015}

\maketitle

\begin{problab}{2.1}
Let $R$ be a domain containing $\C$, and suppose that $R$ is a finite-dimensional $\C$-vector space.  Show that $R=\C$.
\end{problab}

\begin{problab}{2.2}
\begin{enumalph}
\item Compute the minimal polynomial of $\alpha=2\cos(2\pi/5)$.
\item Conclude that the regular $5$-gon is constructible by straightedge and compass, express $\cos(2\pi/5)$ and $\sin(2\pi/5)$ using only square roots, and maybe construct a $5$-gon by hand.
\end{enumalph}
\end{problab}

\begin{problab}{2.3}
Prove that $e=\sum_{n=0}^{\infty} 1/n!$ is irrational, as follows.
\begin{enumalph}
\item Show that $0<N!\cdot(e-s_N)<1/N$ where $s_N=\sum_{n=0}^{N} 1/n!$.
\item Suppose that $e=p/q \in \Q$, and using (a) derive a contradiction.
\end{enumalph}
(It is much harder to show that $e$ is transcendental!)
\end{problab}

\begin{problab}{2.4}
Determine the splitting field and its degree over $\Q$ for the polynomials $X^4+X^2+1$ and $X^5-4$.
\end{problab}

\begin{problab}{2.5}
Let $F$ be a field with $\opchar F \neq 2$, and let $L \supset F$ be a field extension of degree $4$.  
\begin{enumalph}
\item Prove that there exists an intermediate field $L \supsetneq K \supsetneq F$ if and only if $L=F(\alpha)$ with $\alpha$ having minimal polynomial over $F$ of the form $X^4+aX^2+b$ with $a,b \in F$.
\item Suppose that $f(X)=X^4+aX^2+b \in F[X]$ is irreducible in $F[X]$ and that $\sqrt{b} \in F$.  Show that if $\alpha \in L$ is a root of $f$, then so is $\sqrt{b}/\alpha$, and deduce that $L=F(\alpha)$ is a splitting field for $f$.
\end{enumalph}
\end{problab}

\begin{problab}{2.6 (sorta M2-2)}
Let $F$ be a field of characteristic $p$, with $p$ prime, and let $f(X)=X^p-X-a$.  
\begin{enumalph}
\item Show that if $f(X)$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$.
Conclude that if $f(X)$ is irreducible, then $K=F[X]/(f(X))$ is a splitting field for $f$.
\item Conclude that $X^p-X-a$ is irreducible over $\F_p[X]$ for all $a \in \F_p^\times$ and that the fields $K=F[X]/(f(X))$ obtained are isomorphic for all $a$.
\end{enumalph}
\end{problab}

\end{document}