
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 9}

\usepackage{amssymb} 

\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 23 April. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\medskip

\begin{enumerate}

\item  Let $k = {\mathbb Z}/p {\mathbb Z}$, let $k[t]$ be the ring of polynomials in one variable $t$, and let
    $l = k(t)$ be the field of fractions of $k[t]$. Let $f = x^p - t \in l[x]$. 

\begin{enumerate}

\item Prove that $f$ is irreducible in $l[x]$. 

\item Prove that the splitting field of $f$ over $l$ has the form $l(\alpha)$ where $\alpha^p = t$, and describe how
   $f$ splits in $l(\alpha)[x]$. 

\item Prove that $f$ is not separable. 

\end{enumerate} 

\item  Let $k$ be an extension field of $l$, and say that a set $A \subseteq l$ is {\em algebraically independent over $k$}
 if for every distinct $a_1, \ldots a_n$ and every nonzero  $f \in k[x_1, \ldots x_n]$ we have $f(a_1, \ldots a_n) = 0$. 

\begin{enumerate}

\item Prove that there is a maximal algebraically independent set $A \subseteq l$.  

\item Prove that if $A$ is a maximal algebraically independent subset of $l$ then every element of $l$ is algebraic over $k(A)$. 

\item One can show (using properties from HW8 Q1, similar to proofs in linear algebra) that any two maximal algebraically independent sets
   have the same size. We define the {\em transcendence degree of $l$ over $k$} to be the size of a maximal algebraically independent set,
    and refer to such sets as {\em transcendence bases for $l$ over $k$}. 
   Prove that  the transcendence degree of $\mathbb R$ over $\mathbb Q$ is uncountable. 

\end{enumerate}     


\item Prove that if $l$ is an algebraic extension of $k$ and every monic irreducible $f \in k[x]$ has at least one root in $l$, then
   $l$ is algebraically closed. Such extensions of $k$ are called {\em algebraic closures of $k$}. 

\item Fill in the details of the following proof that every field $k$ has an algebraic closure. 

\begin{enumerate} 

\item To every monic irreducible $f \in k[x]$ associate a variable symbol $t_f$, let $R$ be the ring of
    polynomials in the variables $t_f$ with coefficients from $k$, and let $J$ be the ideal 
    of $R$ generated by all the polynomials of the form $f(t_f)$. Then $J \neq R$. 
    
\item There is a maximal ideal $M$ of $R$ with $M \supseteq J$. 

\item The field $l = R/M$ contains an isomorphic copy of $k$, given by $\{ a + M : a \in k \}$.

\item Identifing $a \in k$ with $a + M \in l$, $l$ is an algebraic closure of $k$. 

\end{enumerate}
  

\item  Let $\alpha = 2^{1/3}$ and $\beta = \exp(2 \pi i/3)$, and let $E = {\mathbb Q}(\alpha, \beta)$. As we saw in class, $E$ is a splitting field
     for $x^3 - 2$ over $\mathbb Q$ and $[E : {\mathbb Q}] = 6$. 

\begin{enumerate}

\item Find a basis for $E$ as a VS over $\mathbb Q$.

\item Describe $Aut(E/ {\mathbb Q})$.

\item For each subgroup $H$ of  $Aut(E/ {\mathbb Q})$, compute $Fix(H) = \{ a \in E: \forall \sigma \in H \; \sigma(a) = a \}$. 

\end{enumerate} 

  
\end{enumerate}

\end{document} 
