In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.
Email your librarian or administrator to recommend adding this journal to your organisation's collection.