On embeddings of projective spaces

Abstract

Let k, K be fields and n ≥ 3. Then an n-dimensional projective space Pⁿ(k) can be embedded into a projective plane P₂(K) if K is acyclic extension of k and dim_k K ≥ n + 1. Conversely if an n-dimensional projective space Pⁿ(k) is embedded into a projective plane P² (K) then k is a subfield of K with dim_k K ≥ n + 1.