Semifield planes of order \(p^4\) and kernel \(\mathbf{GF}(p^2)\)
In [\textit{R. Figueroa} and \textit{M. Cordero}, Note Mat. 22, No. 1, 75--81 (2003; Zbl 1093.51009)], the following conjecture was introduced: Let \(p \geq 3\) be prime. If \(\pi\) is a semifield plane of order \(p^4\) and kernel containing \(\text{ GF}(p^2)\), then \(\pi\) is Desarguesian, \(p\)-primitive, or a generalized twisted field plane. In the same paper it was shown that the conjecture is true if \(p=5\). The validity of this conjecture for \(p=3\) was shown in [\textit{V. Boerner-Lantz}, A new class of semifields, PhD Dissertation, Washington State University (1983)]. In the present paper, the author uses a computer to show that the conjecture is also true for \(p=7\) and \(p=11\). In the mean time the conjecture has been shown for any prime \(p \geq 3\) without the use of the computer [see \textit{I. Cardinali, O. Polverino} and \textit{R. Trombetti}, Eur. J. Comb. 27, No. 6, 940--961 (2006; Zbl 1108.51010)].
scientific article; zbMATH DE number 5000035