\centerline{Exercises VIII}

\bigskip\noindent
{\bf 1.} A {\it derivation} of a ring $R$ is a map $D : R \to R$ such
that $D(x+y)=D(x)+D(y)$ and $D(xy)=xD(y)+D(x)y$. Note that the zero
map $D(x)=0$ is a derivation of any ring.
\item{(a)} Show that if $R={\bf Z}$, $R={\bf Q}$, or $R={\bf Z}/n{\bf Z}$ 
then the only derivation of $R$ is the zero derivation. (Hint: show that 
$D(1)=0$ for any derivation of any ring.)
\item{(b)} Show that any derivation of ${\bf Z}[x]$ is of the form 
$D=g{\rm d}/{\rm d}x$ for some $g\in {\bf Z}[x]$.
\item{(c)} Let $f\in {\bf Q}[x]$ be a polynomial and let $R={\bf Q}[x]/(f)$. 
For which $f$ does $R$ have a nonzero derivation?

\medskip\noindent
{\bf 2.} Let $K=F(\alpha)/F$ be a primitive algebraic extension of
fields of characteristic $p$. A {\it derivation of} $K$ {\it over} $F$
is a derivation $D$ of $K$ such that $D(a)=0$ for all $a\in F$.  We
use the notation ${\rm Der}(K/F)$ to denote the set of derivations of
$K$ over $F$.
\item{(a)} When does $K$ have a nonzero derivation $D$ over $F$?
\item{(b)} Consider the subfield $F \subset L \subset K$ defined by the 
rule
$$
L = \{ x\in K\ \ | \ \ D(x)=0, \forall D\in {\rm Der}(K/F) \}.
$$
Characterize $L$ in terms of $\alpha$ and $Irr(\alpha, F)$.

\medskip\noindent
{\bf 3.} Prove that
$$
{\bf Q} = {\bf Q}( 
(2 + {10 \over 9}\sqrt{3})^{1/3} + (2 - {10 \over 9}\sqrt{3})^{1/3} 
).
$$

\medskip\noindent
{\bf 4.} Show that in a finite field every element is the sum of two 
squares.

\medskip\noindent
{\bf 5.} Let $F$ be a field of characteristic $p>0$, and let $a\in F$.
Show that $X^p-X+a$ either splits completely over $F$ or is irreducible
over $F$.

\medskip\noindent
{\bf 6.}  What is the Galois group of the polynomial $ 3x^6 - 12x^4 +
4x^3 + 12x^2 - 8 $ over ${\bf Q}$?  (This should be pretty tough! Feel
free to use computer algebra to find the answer, but try to write up
the arguments by hand. For example, write the roots as $\alpha_1,
\ldots, \alpha_6$ and try to find expressions in them which are in
${\bf Q}$ to restrict the size of the Galois group.)

\bye
