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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
##### Keywords:
semifields; semifield planes; finite affine planes
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