Exercise 21, § 11

(a) Show directly (without using the isomorphisms $\phi_p$) that every $(n - 1)$-vector over an n-dimensional vector space is pure.

(b) Let A be a commutative algebra over a field K with a basis consisting of the unit element 1 and three elements $a_1, a_2, a_3$ whose products with one another are zero. Let E be the A-module $A^3$ and $(e_i)_{1 \leq i \leq 3}$ its canonical basis; show that the bivector

is not pure.