%%This is a standard LaTeX2e article document template. personal version 12/5/200%% \documentclass[11pt,twoside]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%packages%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{empty} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amstext} \usepackage{multicol} \usepackage[all]{xy} \usepackage{amsxtra} \usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref} \hypersetup{ pdftitle = {Math 371 Assignment 7}, pdfauthor = {\textcopyright\ Bryden Cais}, pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq}, colorlinks = {true} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%formatting%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\topmargin}{0in} %%% This sets all the spacing stuff to use the page more \setlength{\oddsidemargin}{0in} %%% efficiently than the normal "article" setup would. \setlength{\evensidemargin}{0in} %%% It's OK to play with these some. \setlength{\textheight}{8.5in} %%% \setlength{\textwidth}{6.5in} %%% \setlength{\headsep}{0in} %%% \setlength{\headheight}{0in} %%% %\setlength{\footskip}{0in} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \newcommand{\Q}{\mathbf{Q}} \newcommand{\R}{\mathbf{R}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\mathbf{F}} \newcommand{\p}{\mathfrak{p}} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \begin{document} \begin{center} {\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ {\large Assignment 7}\\Due Friday, April 9 at 08:35 \end{center} \begin{enumerate} \item Let $p$ be a prime and let $K$ be a splitting field of $X^p-2\in \Q[X]$, so $K/\Q$ is a Galois extension. Show that $K=\Q(a,\zeta)$ for $a\in K$ satisfying $a^p=2$ and $\zeta\in K$ a primitive $p$ th root of unity. Describe generators of $G:=\Gal(K/\Q)$ in terms of their actions on $a$ and $\zeta$, and describe $G$ as an abstract group (in terms of generators and relations, say). Write out the diagrams of intermediate fields and groups, indicating clearly the various containments. Also indicate which subfields of $K$ are Galois over $\Q$. \item Let $F$ be a finite field of size $\#F$, with $K/F$ a finite extension of degree $d$. Prove that $K/F$ is Galois and that $\Gal(K/F)$ is a cyclic group of order $d$ with generator the automorphism of $K$ given by $$\alpha\mapsto \alpha^{\#F}.$$ (This automorphism is called the {\em arithmetic Frobenius} map of $F$). \item This exercise gives Artin's proof of the fundamental theorem of Algebra. Let $F$ be a field not of characteristic 2 and assume that all odd degree polynomials in $F[X]$ have a root in $F$. Let $K$ be a quadratic extension of $F$ with the property that every element of $K$ has a square root in $K$. \begin{enumerate} \item Prove that any finite extension of $K$ has degree a power of $2$. (Hint: Reduce to the Galois case and then consider the fixed field of the 2-Sylow subgroup of the Galois group). \item Prove that $K$ has no non-trivial finite extensions which are Galois over $F$, and conclude that $K$ is algebraically closed. (Hint: Use the fact that a non-trivial 2-group has an index 2 normal subgroup). \item Let $F=\R$ and $K=\R[X]/(X^2+1)$. Explain (using the intermediate value theorem) why $F$ satisfies the hypotheses above, and using explicit formulae, show that $K$ also satisfies the hypotheses. Conclude that $\C:=K$ is algebraically closed (this is the Fundamental Theorem of Algebra). \end{enumerate} \item Determine the Galois group of the splitting field (over $\Q$) of $X^4-14X^2+9$, and write down the lattice of subgroups and corresponding subfields. Which subfields are Galois over $\Q$? \item Fix a positive integer $n$ and let $K:=\Q(\zeta_n)$ for a primitive $n$ th root of unity $\zeta_n\in \C$. Prove that complex conjugation $\tau\in \Aut(\C)$ restricts to an automorphism of $K$ fixing $\Q$, and show that the corresponding element of $\Gal(K/\Q)$ corresponds to $-1$ under the isomorphism $\Gal(K/\Q)\simeq (\Z/n\Z)^{\times}$. Prove that the fixed field $K^+$ of the subgroup generated by complex conjugation is equal to the intersection $K\cap \R$ taken inside $\C$. We call $K^+$ the {\em maximal real subfield} of $K$. \item This problem works out a formula for $\cos(2\pi/17)$ in terms of square-root extractions. Let $\zeta:=e^{2\pi i/17}$; it is a primitive $17$ th root of unity. Let $\alpha:=\zeta+\zeta^{-1}=2\cos(2\pi/17)$. Let $\sigma\in \Gal(\Q(\zeta)/\Q)$ be the element determined by $$\sigma \zeta = \zeta^3.$$ \begin{enumerate} \item Show that $\sigma$ generates $\Gal(\Q(\zeta)/\Q)$ \item Define the {\em periods} of $\alpha$ to be \begin{align*} & \eta_1:= \alpha + \sigma^2\alpha + \sigma^4\alpha + \sigma^8\alpha && & \eta_1':=\sigma\eta_1\\ & \eta_2 := \alpha + \sigma^4\alpha && & \eta_2':=\sigma^2 \eta_2 \\ & \eta_3 := \sigma \eta_2 && & \eta_3':=\sigma\eta_2' \end{align*} Prove that $\eta_1,\eta_1'$ are the roots of $X^2+X-4$, that $\eta_2,\eta_2'$ are the roots of $X^2-\eta_1X-1$, that $\eta_3,\eta_3'$ are the roots of $X^2-\eta_1'-1$ and that $\alpha$ and $\sigma^4\alpha$ are the roots of $X^2-\eta_2X+\eta_3$. \item Conclude that $\cos(2\pi/17)$ is equal to \begin{equation*} \frac{1}{16}\left(-1 +\sqrt{17}+\sqrt{2(17-\sqrt{17})} +2\sqrt{17+3\sqrt{17}-\sqrt{2(17-\sqrt{17})}-2\sqrt{2(17+\sqrt{17})}}\right) \end{equation*} \end{enumerate} \item Let $F$ be a field and $f\in F[X]$ a monic separable polynomial of degree $n$. Fix a splitting field $K$ of $f$ and write $G:=\Gal(K/F)$. \begin{enumerate} \item Prove that $G$ is a subgroup of $S_n$, the symmetric group on $n$ letters. If $f$ is irreducible, prove that $G$ is a {\em transitive} subgroup of $S_n$ with $\#G$ divisible by $n$. (Viewing $S_n$ as the permutations of an $n$-element set $T$, a transitive subgroup $G$ is one which acts transitively on these $n$ elements, i.e. for any $x,y\in T$ there exists $g\in G$ such that $gx=y$). \item Prove that if $n$ is prime and $f$ is irreducible, then $G$ contains an $n$-cycle. (Hint: use Sylow's theorem.) \label{ncycle} \item Suppose that $f$ is irreducible of degree $5$ and has exactly 3 real roots. Prove that $G$ is isomorphic to $S_5$. (Hint: View $K$ as a subfield of $\C$ and consider complex conjugation acting on $K$. Now use (\ref{ncycle}) and some group theory.) \end{enumerate} \item Keep the notation of the previous problem. \begin{enumerate} \item Let $r_1,\ldots,r_n$ be the $n$ distinct roots of $f$ in $K$, and define the {\em discriminant of $f$} to be $$\Delta(f):= \prod_{i,j} (r_i-r_j),$$ where the product runs over all pairs $(i,j)\in \Z^2$ with $1\le i,j\le n$. Prove that $\Delta(f)\in F$. \item Prove that $G$ is a subgroup of $A_n$ (the alternating group) if and only if $\Delta(f)$ is a square in $F$. Hint: use the formula for $\Delta(f)$ above and the definition of $A_n$ as the group of even permutations. \end{enumerate} \end{enumerate} \end{document}