
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 8}

\usepackage{amssymb} 

\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 16 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 $k$ be a subfield of $l$. We write $P(l)$ for the set of subsets of $l$,
  and define a map $acl$ from $P(l)$ to $P(l)$ as follows: $acl(X)$ is the set
  of $b \in l$ such that $b$ is algebraic over $k(X)$.  Prove that:

  \begin{enumerate}

  \item  $X \subseteq acl(X)$. 
    
  \item If $X \subseteq Y$ then $acl(X) \subseteq acl(Y)$.

  \item $acl(acl(X)) = acl(X)$.

  \item If $b \in acl(X)$ then there is finite $x \subseteq X$ such that $b \in acl(x)$.

  \item If $b \in acl(X \cup \{a\}) \setminus acl(X)$, then $a \in acl(X \cup \{ b \})$.

  \end{enumerate} 

  What do these properties remind you of? 


\item Let $k$ be a subfield of $l$ and let $a, b \in l \setminus k$ be such that $[k(a): k]$ and
     $[k(b): k]$ are coprime. Prove that $m_a^k = m_a^{k(b)}$. 


\item Let $p$ be a prime and let $k$ be a field of characteristic $p$. Define $\phi: k \rightarrow k$ by
  $\phi: a \mapsto a^p$. Prove that:

\begin{enumerate} 

\item The map $\phi$ is a monomorphism from $k$ to $k$.

\item If $k$ is finite then $\phi$ is an automorphism of $k$.

\item If $k$ is finite and $\vert k \vert = p^n$ then $\phi$ is an element of order $n$
    in the group $Aut(k)$. 

\end{enumerate}

\item  Let $k = {\mathbb Z}/2 {\mathbb Z}$. Find an irreducible $f \in k[x]$ of degree three.
  Now let $l = k[x]/(f)$, let $\alpha = x + (f)$ and identify $a \in k$ with
  $a + (f) \in l$.

  \begin{enumerate}

  \item  Show that $l = \{ a + b\alpha + c \alpha^2 : a, b, c \in k \}$.

  \item  Find the minimal polynomial of each element in $l$ over $k$.

  \item  Compute the group $Aut(l/k)$.

  \end{enumerate}


\item  Let $k$ be a subfield of $l$ and let $a \in l$ be transcendental over $k$.

  \begin{enumerate}

  \item Prove that $k(a)$ is isomorphic to $k(x)$ (the field of fractions of $k[x]$).

  \item Prove that $k(a)$ is not finitely generated as a ring over $k$, that is to say there do not exist
    elements $b_1, \ldots b_n \in k(a)$ such that $k(a) = k[b_1, \ldots b_n]$. Hint: There are infinitely many
    irreducibles in $k[x]$.

  \end{enumerate}   

\item Use the Eisenstein criterion to prove that for any prime $p$, the polynomial
     $\frac{ (x+1)^p - 1}{x}$ is irreducible. 
  
\item Let $p > 2$ be prime and let $\beta = \exp(2 \pi i/p)$, so that $\beta$ is an element of order
  $p$ in the multiplicative group of the complex numbers. Find the minimal polynomial of $\beta$ over $\mathbb Q$. 

  

  
  
\end{enumerate}

\end{document} 
