A k-field is a field over which
every polynomial of degree less than or equal to k splits completely. The main
theorem characterizes the maximal decomposable subspaces of the k-th symmetric
space ∨kV , where V is finite-dimensional vector space over an infinite k-field.
They come in three forms: (1) {x1∨⋯∨ xk: xk∈ V },x1,⋯,xk−1 fixed; (2)
⟨a,b⟩k= {x1∨⋯∨ xk: xi∈⟨a,b⟩}; and (3) {x1∨⋯∨ xk−r∨⟨a,b⟩(r′)},x1,⋯ , Xk-r
fixed; where a and b are linearly independent vectors in V and ⟨a,b⟩ is the subspace
spanned by a and b.
