k-invariant nets over an algebraic extension of a field k

Abstract

Let K be an algebraic extension of a field k, let σ = (σ_ij) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σ_ij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ⊆ P ⊆ K, up to conjugation by a diagonal matrix.