Geodesic
Minimal polynomials in finite semifields
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 5, pp. 588-596.

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We consider the classical notion of a minimal polynomial and apply it to investigations in finite semifields. A proper finite semifield has non-associative multiplication, that leads to a number of anomalous properties of one-side-ordered minimal polynomials. The interrelation between the minimal polynomial of an element and the minimal polynomial of its matrix from the spread set is described and illustrated by some semifields of orders 16, 32 and 64.
Keywords: semifield, right-ordered degree, right-ordered minimal polynomial. 
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Olga V. Kravtsova. Minimal polynomials in finite semifields. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 5, pp. 588-596. http://geodesic.mathdoc.fr/item/JSFU_2018_11_5_a5/

[1] L.E. Dickson, “Linear algebras in which division is always uniquely possible”, Trans. Amer. Math. Soc., 7 (1906), 370–390 | DOI | MR | Zbl

[2] A.A. Albert, “Finite division algebras and finite planes”, Proc. Sympos. Appl. Math., 10, AMS, Provid., R.I., 1960, 53–70 | DOI | MR

[3] N.L. Johnson, V. Jha, M. Biliotti, Handbook of finite translation planes, Pure and applied mathematics, Chapman/CPC, 2007 | MR | Zbl

[4] V.M. Levchuk, O.V. Kravtsova, “Problems on structure of finite quasifields and projective translation planes”, Lobachevskii Journal of Mathematics, 38:4 (2017), 688–698 | DOI | MR | Zbl

[5] D.R. Hughes, F.C. Piper, Projective planes, Springer-Verlag New-York Inc., 1973 | MR | Zbl

[6] R. Lidl, G. Pilz, Applied Abstract Algebra, Springer-Verlag, New York, 1984 | MR | Zbl

[7] G.P. Wene, “On the multiplicative structure of finite division rings”, Aequationes Math., 41 (1991), 791–803 | DOI | MR

[8] I.F.Rúa, “Primitive and Non-Primitive Finite Semifields”, Commun. Algebra, 22 (2004), 223–233 | MR

[9] I.R. Hentzel, I.F. Rúa, “Primitivity of Finite Semifields with 64 and 81 elements”, International Journal of Algebra and Computation, 17:7 (2007), 1411–1429 | DOI | MR | Zbl