Some examples of denumerable pseudo-ordered fields and their applications to geometry

• Autorzy: Jakóbowski J.
• Języki publikacji: EN

Abstrakty

EN

The paper deals with the following problems concerning pseudo-ordered denumerable fields: (i) a connection between Fermat’s two squares theorem and the unique pseudo-order in a finite field; (ii) properties of a proper pseudo-order determined by any prime number in the field of rational numbers; (iii) existence of a proper pseudo-order in every subfield of the sequence used to obtain the field of constructible numbers; (iv) some brief of applications of the latter pseudo-orders to construct new algebraic and geometric structures. In particular, we extend the known construction of finite nearfields or quasifields given by e.g. W. A. Pierce or P. Dembowski – to infinite cases.

Słowa kluczowe

EN

Fermat's two squares theorem; Minkowski plane; pseudo-ordered fields; geometry

PL

twierdzenie dwóch kwadratów Fermata; płaszczyzna Minkowskiego; pseudo-uporządkowane pole; geometria

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 2
• Strony: 247--256
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Jakóbowski J.

• University of Warmia and Mazury in Olsztyn, Faculty of Mathematics and Computer Science, 10-710 Olsztyn, Poland

Bibliografia

• [1] L. Carlitz, A theorem on permutations in a finite fields, Proc. Amer. Math. Soc. 11 (1966), 456–459.
• [2] P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-Heidelberg-New York, 1968.
• [3] J. A. Ewell, A simple proof of Fermat’s two squares theorem, Amer. Math. Monthly 90 (1983), 635–637.
• [4] J. Jakóbowski, A new construction for Minkowski planes, Geom. Dedicata 69 (1998), 179–188.
• [5] J. Jakóbowski, A new generalization of Moulton affine planes, Geom. Dedicata 42 (1992), 243–253.
• [6] J. Jakóbowski, Nearaffine planes related to pseudo-ordered fields, Bull. Polish Acad. Sci. Math. 50 (2002), 345–360.
• [7] S. Lang, Algebra, Addison-Wesley Publishing Company, 1970.
• [8] N. Percsy, Finite Minkowski planes in which every circle-symmetry is an automorphism, Geom. Dedicata 10 (1981), 269–282.
• [9] W. A. Pierce, Moulton planes, Canad. J. Math. 13 (1961), 427–436.
• [10] C. R. Videla On the constructible numbers, Proc. Amer. Math. Soc. 127 (1999), 851–860.
• [11] H. A. Wilbrink, Nearaffine planes, Geom. Dedicata 12 (1982), 53–62.
• [12] H. A. Wilbrink, Finite Minkowski planes, Geom. Dedicata 12 (1982), 119–129.