Exercise 7, § 10

a) Show that the following properties of $k$ are equivalent:

(i) Every anisotropic semi-simple $k$-Lie algebra reduces to 0.

(ii) Every semi-simple $k$-Lie algebra has a Borel subalgebra.

(Use Exerc. 6 to prove that (i) $=\Rightarrow$ (ii).)

b) Show that (i) and (ii) imply$^{19}:$

(iii) Every finite dimensional $k$-algebra that is a field is commutative. (Or again: the Brauer group of every algebraic extension of $k$ reduces to 0.)

(Use the Lie algebra of elements of trace zero in such an algebra.)

c) Show that (i) and (ii) are implied by:

(iv) For any finite family of homogeneous polynomials $f_{\alpha}\in k[(X_i)_{i\in I}]$ of degrees $\geq 1$ such that $\sum_{\alpha}$ deg $f_{\alpha}<$ Card(I), there exist elements $x_i\in k$, not

all zero, such that $f_{\alpha}((x_i)_{i\in I}) = 0$ for all $\alpha$.

(Use Prop. 5 of §8.)