Exercise 13, § 2

Assume that $k$ is finite. Denote by $k'$ an infinite extension of $k$. Let $\mathfrak{g}$ be a Lie algebra over $k$. The rank of $\mathfrak{g}$, denoted by rk($\mathfrak{g}$), is the rank of the $k'$-Lie algebra $\mathfrak{g}'=\mathfrak{g}\otimes_kk'$; an element of $\mathfrak{g}$ is said to be regular if it is regular in $\mathfrak{g}'$; these definitions do not depend on the choice of $k'$. Show that, if

Card($k$)$\geq$ dim$\mathfrak{g}-$ rk($\mathfrak{g}$),

$\mathfrak{g}$ contains a regular element (hence also a Cartan subalgebra).

(Use the following result: if $a$ is a non-zero homogeneous element of $k[X_1, . . . ,X_n]$, and if Card($k$)$\geq$ deg($a$), there exists $x\in k^n$ such that $a(x)\not= 0.$)