Czasopismo
Tytuł artykułu
Warianty tytułu
Języki publikacji
Abstrakty
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.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
333-349
Opis fizyczny
Daty
wydano
2015-12-01
otrzymano
2015-08-14
online
2016-03-25
Twórcy
autor
- Institute of Computer Science, University of Gdańsk, Poland
autor
- 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