In mathematics, André planes are a class of finite translation planes found by André. The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular nearfield planes are also André planes.
Construction.
Let formula_1 be a finite field, and let formula_2 be a degree formula_3 extension field of formula_4. Let formula_5 be the group of field automorphisms of formula_6 over formula_4, and let formula_8 be an arbitrary mapping from formula_4 to formula_5 such that formula_11. Finally, let formula_12 be the norm function from formula_6 to formula_4.
Define a quasifield formula_15 with the same elements and addition as K, but with multiplication defined via formula_16, where formula_17 denotes the normal field multiplication in formula_6. Using this quasifield to construct a plane yields an André plane.
Small Examples.
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:
References.
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