Exercise 23, § 1

Let $g$ be a finite-dimensional commutative Lie $p$-algebra over a perfect field $K$ of characteristic $p > 0$.

(a) Show that $g$ is in a unique way the direct sum of two $p$-subalgebras $h, t$ such that on $h$ the $p$-mapping is bijective and on $t$ it is nilpotent (consider the iterations of the $p$-mapping on $g$; cf. Algebra, Chapter VIII, § 2, no. 2, Lemma 2 and Chapter VII, § 5, Exercise 20 (a)). $h$ is called the $p$-core of $g$.

(b) Suppose that $K$ is algebraically closed. Show that there exists a basis $(e_i)$ of $h$ such that $e_i^p = e_i$ for all $i$ (consider a $p$-subalgebra of $h$ generated by a single element and show that it contains some $x$ such that $x^p = x \neq 0$; then argue by induction on the dimension of $h$). Deduce that the $p$-subalgebras of $h$ are finite in number and are in one-to-one correspondence with the vector subspaces of the vector space over the prime field $\mathbf{F}_p$ generated by the $e_i$.

(c) Show that there exists a basis $(f_{ij})$ of $t$ ($1 \leqslant i \leqslant k, 1 \leqslant j \leqslant s_i$ for all $i$) where the sequence of $s_i$ is decreasing and which is such that $f_{1j}^p = 0$ for $1 \leqslant j \leqslant s_1, f_{1j}^p = f_{i-1,j}$ for $2 \leqslant i \leqslant k, 1 \leqslant j \leqslant s_i$ (method of Algebra, Chapter VII, § 5, Exercise 20 (b)).