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.
Let be a finite field, and let be a degree extension field of . Let be the group of field automorphisms of over , and let be an arbitrary mapping from to such that . Finally, let be the norm function from to .
Define a quasifield with the same elements and addition as K, but with multiplication defined via , where 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]
Order | Number of
non-Desarguesian Andrè planes |
---|---|
9 | 1 |
16 | 1 |
25 | 3 |
27 | 1 |
49 | 7 |
64 | 6 (four 2-d, two 3-d) |
81 | 14 (13 2-d, one 4-d) |
121 | 43 |
125 | 6 |
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.
Let be a finite field, and let be a degree extension field of . Let be the group of field automorphisms of over , and let be an arbitrary mapping from to such that . Finally, let be the norm function from to .
Define a quasifield with the same elements and addition as K, but with multiplication defined via , where 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]
Order | Number of
non-Desarguesian Andrè planes |
---|---|
9 | 1 |
16 | 1 |
25 | 3 |
27 | 1 |
49 | 7 |
64 | 6 (four 2-d, two 3-d) |
81 | 14 (13 2-d, one 4-d) |
121 | 43 |
125 | 6 |