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Tytuł artykułu

Characteristic of Rings. Prime Fields

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.
Twórcy
  • Institute of Computer Science, University of Gdańsk, Poland
  • Institute of Informatics, University of Białystok, Poland
Bibliografia
  • [1] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.
  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
  • [5] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.
  • [6] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.
  • [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.
  • [8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.
  • [9] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.
  • [10] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
  • [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.
  • [12] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Set of points on elliptic curve in projective coordinates. Formalized Mathematics, 19(3):131–138, 2011. doi:10.2478/v10037-011-0021-6.[Crossref]
  • [13] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers. Formalized Mathematics, 21(2):115–125, 2013. doi:10.2478/forma-2013-0013.[Crossref]
  • [14] Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009.
  • [15] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.
  • [16] Artur Korniłowicz and Christoph Schwarzweller. The first isomorphism theorem and other properties of rings. Formalized Mathematics, 22(4):291–301, 2014. doi:10.2478/forma-2014-0029.[Crossref]
  • [17] Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59–68, 2003.
  • [18] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.
  • [19] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.
  • [20] Heinz Lüneburg. Die grundlegenden Strukturen der Algebra (in German). Oldenbourg Wisenschaftsverlag, 1999.
  • [21] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265–269, 2001.
  • [22] Michał Muzalewski. Opposite rings, modules and their morphisms. Formalized Mathematics, 3(1):57–65, 1992.
  • [23] Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643–648, 1991.
  • [24] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.
  • [25] Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.
  • [26] Karol Pąk. Linear map of matrices. Formalized Mathematics, 16(3):269–275, 2008. doi:10.2478/v10037-008-0032-0.[Crossref]
  • [27] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.
  • [28] Christoph Schwarzweller. The correctness of the generic algorithms of Brown and Henrici concerning addition and multiplication in fraction fields. Formalized Mathematics, 6(3): 381–388, 1997.
  • [29] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.
  • [30] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69–79, 1998.
  • [31] Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115–122, 2008. doi:10.2478/v10037-008-0017-z.[Crossref]
  • [32] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.
  • [33] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.
  • [34] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.
  • [35] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
  • [36] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.
  • [37] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.
  • [38] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
  • [39] B.L. van der Waerden. Algebra I. 4th edition. Springer, 2003.
  • [40] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
bwmeta1.id-class.MML
RING 3
Identyfikator YADDA
bwmeta1.element.doi-10_1515_forma-2015-0027
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