\documentclass{article}
\pagestyle{empty}
\usepackage{amssymb}
\usepackage{latexsym}
 
\title{MATH0037 Galois Theory 2002, Sheet 6}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
\begin{document}
\maketitle
 \begin{enumerate}
\item Suppose that $K$ is a field. Define a map
$D_K : K[X] \rightarrow K[X]$ by
\[\sum_{i=0}^n a_i X^i \mapsto \sum_{i=0}^n ia_iX^{i-1}.\]
This map is familiar in the context that
$K = \mathbb R$, but that is not our assumption.
We may write $f'$ for $D_K(f)$.
\begin{enumerate}
\item  The ring $K[X]$ is vector space over $K$. Show
that the map $D_K$ is a linear map.
\item  Suppose that $f, h \in K[X]$. Prove that
$D_K(fh) = D_K(f)h + fD_K(h).$
\item Suppose that $K$ is a subfield of $L$. Prove that
if $f \in K[X],$ then $D_K(f) = D_L(f)$.
\item Suppose that $f \in \mathbb Q[X]$ is irreducible.
Show that there is no field $L$ with subfield $\mathbb Q$
such that there is $\alpha \in L$ with $f = [X-\alpha]^2 h$
for some polynomial $h \in L[X]$.
\item$Harder$ Let $F=\mathbb Z_p$ be the field which
consists of the integers modulo
$p$. The polynomial ring $F[Y]$ is an integral domain so
we may form its field of fractions $K$ by a construction
found on problem sheet 3 (Question 1). Now consider
$X^p - Y \in K[X]$. Show that in $K[X]$ this polynomial
is irreducible, but that given any field $L$ with 
$K < L$ such that $L$ contains a root $\alpha$ of $X^p-Y$,
then $X^p-Y = [X-\alpha]^2 h$ for some polynomial $h \in L[X]$.  
\end{enumerate}
\item Suppose that $K$ is a field and that
$0 \not = f \in K[X]$. Now $R = K[X]/(f)$ has a natural
vector space structure. Prove that dim$_K R = \mbox{deg }f$.
What happens if $f = 0$?
\item Suppose that $K$ is a field. Suppose that $f, h \in K[X] = R$
and that neither $f$ nor $h$ is the zero polynomial.
We have two natural maps $\pi_1 : R 
\mapsto K[X]/(f) = R_1$ and $\pi_2 : K[X] \mapsto
K[X]/(h) 
= R_2$. Define a map
$\pi : R \mapsto R_1 \oplus R_2$ by
$g \mapsto (\pi_1(g), \pi_2(g))$ for all $g \in R$.
Suppose that $(f) + (h) = R$.
\begin{enumerate}
\item Prove that
$\pi$ is a surjective ring homomorphism and
a linear map between $K$-vector spaces.
\item Prove that Ker $\pi = (f) \cap (h) = (fh)$.
\item By considering dimensions, prove that
$\pi$ is surjective.
\end{enumerate}
\end{enumerate}
\end{document}
