Exercise 21, § 1

(a) Let $K$ be a field of characteristic $p > 0$ and $L$ a not necessarily associative algebra over $K$. Show that the derivations of $L$ form a Lie $p$-algebra with the $p$-mapping $D \mapsto D^p$.

(b) Let $L$ be the algebra $K[X]/(X^p)$ and $w$ the $p$-algebra of derivations of $L$. Let $D_i$ ($0 \leq i \leq p-1$) denote the derivation of $L$ such that $D_i(X) = X^i$. Show that the $D_i$ form a basis of $w$ over $K$ and that $[D_i, D_j] = (j - i)D_{i+j-1}$ if $i + j \leq p$, $[D_i, D_j] = 0$ if $i + j > p$, and $D_i^p = 0$, except for $i = 1$, in which case $D_1^p = D_1$.

(c) Show that, if $p \geq 3$, the Lie algebra $w$ only admits the ideals $\{0\}$ and $w$. (Using the multiplication table of the $D_i$, show that every non-zero ideal of $w$ contains a non-zero multiple of $D_{p-1}$.)

(d) Let $V_k$ be the subspace of $w$ with basis the $D_i$ of index $\geq k$. Show that, if $p \geqslant 5$, $V_k$ is identical with the set of $Z \in w$ whose centralizer is of dimension $\geqslant k + 1$. Deduce that, if $p \geqslant 5$, the automorphism group of $w$ is solvable.