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On the semifield planes of order \(5^4\) and dimension 2 over the kernel. (English) Zbl 1093.51009
The authors conjecture that there are exactly \(1+(p-2)+((p+1)/2)^2\) non isomorphic semifield planes of order \(p^4\) and kernel containing GF\((p^2)\) for each prime number \(p > 2\). Of these, one is the desarguesian plane, \(p-2\) are generalized twisted field planes and \(((p+1)/2)^2\) are \(p\)-primitive planes. They construct all the semifield planes of order \(5^4\) and kernel containing GF\((5^2)\) and prove the conjecture in the case \(p = 5\). The conjecture was proved for the case \(p = 3\) in [V. Boerner-Lantz, A new class of semifields, PhD Dissertation, Washington State University (1983)].

MSC:
51E15 Finite affine and projective planes (geometric aspects)
51A40 Translation planes and spreads in linear incidence geometry
05B25 Combinatorial aspects of finite geometries
17A35 Nonassociative division algebras
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