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Commutative presemifields and semifields. (English) Zbl 1194.12007
Summary: Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely, any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert’s commutative twisted fields. A classification of all planar Dembowski-Ostrom polynomials over any finite field of order \(p^3\), \(p\) an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials \(X^{10}+aX^6 - a^2X^2\) with \(a\neq 0\) describes precisely two new commutative presemifields of order \(3^e\) for each odd \(e\geq \)5.

MSC:
12K10 Semifields
12E20 Finite fields (field-theoretic aspects)
17A35 Nonassociative division algebras
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry
Software:
Magma
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References:
[1] Albert, A.A., On nonassociative division algebras, Trans. amer. math. soc., 72, 296-309, (1952) · Zbl 0046.03601
[2] Albert, A.A., Finite division algebras and finite planes, (), 53-70
[3] Albert, A.A., Generalized twisted fields, Pacific J. math., 11, 1-8, (1961) · Zbl 0154.27203
[4] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system I: the user language, J. symbolic comput., 24, 235-265, (1997) · Zbl 0898.68039
[5] Cohen, S.D.; Ganley, M.J., Commutative semifields, two-dimensional over their middle nuclei, J. algebra, 75, 373-385, (1982) · Zbl 0499.12021
[6] Coulter, R.S., The classification of planar monomials over fields of prime square order, Proc. amer. math. soc., 134, 3373-3378, (2006) · Zbl 1122.11083
[7] Coulter, R.S.; Matthews, R.W., Planar functions and planes of Lenz-Barlotti class II, Des. codes cryptogr., 10, 167-184, (1997) · Zbl 0872.51007
[8] Dembowski, P.; Ostrom, T.G., Planes of order n with collineation groups of order n2, Math. Z., 103, 239-258, (1968) · Zbl 0163.42402
[9] Dickson, L.E., On commutative linear algebras in which division is always uniquely possible, Trans. amer. math. soc., 7, 514-522, (1906) · JFM 37.0112.01
[10] Ding, C.; Yuan, J., A family of skew Hadamard difference sets, J. combin. theory. ser. A, 113, 1526-1535, (2006) · Zbl 1106.05016
[11] Ganley, M.J., Central weak nucleus semifields, European J. combin., 2, 339-347, (1981) · Zbl 0469.51005
[12] Gluck, D., A note on permutation polynomials and finite geometries, Discrete math., 80, 97-100, (1990) · Zbl 0699.51008
[13] Hiramine, Y., A conjecture on affine planes of prime order, J. combin. theory ser. A, 52, 44-50, (1989) · Zbl 0696.51004
[14] Hiramine, Y., On planar functions, J. algebra, 133, 103-110, (1990) · Zbl 0702.20016
[15] Kantor, W.M., Commutative semifields and symplectic spreads, J. algebra, 270, 96-114, (2003) · Zbl 1041.51002
[16] Knuth, D.E., Finite semifields and projective planes, J. algebra, 2, 182-217, (1965) · Zbl 0128.25604
[17] Lidl, R.; Mullen, G.L.; Turnwald, G., Dickson polynomials, Pitman monogr. surv. pure appl. math., vol. 65, (1993), Longman Scientific and Technical Essex, England · Zbl 0823.11070
[18] Menichetti, G., On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. algebra, 47, 400-410, (1977) · Zbl 0362.17002
[19] Moore, E.H., A doubly-infinite system of simple groups, Bull. New York math. soc., 3, 69-82, (1893) · JFM 25.0198.01
[20] Moore, E.H., A doubly-infinite system of simple groups, (), 208-242 · JFM 25.0198.01
[21] Penttila, T.; Williams, B., Ovoids of parabolic spaces, Geom. dedicata, 82, 1-19, (2000) · Zbl 0969.51008
[22] Rónyai, L.; Szőnyi, T., Planar functions over finite fields, Combinatorica, 9, 315-320, (1989) · Zbl 0692.05014
[23] Wedderburn, J.H.M., A theorem on finite algebras, Trans. amer. math. soc., 6, 349-352, (1905) · JFM 36.0139.01
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