In mathematics, André planes are a class of finite translation planes found by André.
[1] The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular nearfield planes are also André planes.
be a finite field, and let
extension field of
be the group of field automorphisms of
be an arbitrary mapping from
be the norm function from
Define a quasifield
with the same elements and addition as K, but with multiplication defined via
denotes the normal field multiplication in
Using this quasifield to construct a plane yields an André plane.
[2] For planes of order 25 and below, classification of Andrè planes is a consequence of either theoretical calculations or computer searches which have determined all translation planes of a given order: Enumeration of Andrè planes specifically has been performed for other small orders:[7] non-Desarguesian Andrè planes