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## Demonstratio Mathematica

Tytuł artykułu

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

Autorzy Jakóbowski, J.
Treść / Zawartość https://www.degruyter.com/view/j/dema
Warianty tytułu
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
 PL twierdzenie dwóch kwadratów Fermata   płaszczyzna Minkowskiego   pseudo-uporządkowane pole   geometria EN Fermat's two squares theorem   Minkowski plane   pseudo-ordered fields   geometry
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, jjakob@matman.uwm.edu.pl
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.
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