The six semifield planes associated with a semifield flock
- 作者：Ball ; Simeon ; Brown ; Matthew R.
- 关键词：12K10 ; 51E15 ; 51A40 ; Semifields ; Division rings ; Translation planes ; Flocks
- 刊名：Advances in Mathematics
- 出版年：2004
- 期刊代码：62_00018708
- 类别：mt
- 出版时间：December 1, 2004
- 卷：189
- 期：1
- 页码：68-87
- 文件大小：293 K
摘要
In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (aijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)2+4xf(x) is a non-square for all x
GF(q)*, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson–Kantor–Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.
GF(q)*, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson–Kantor–Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.