Characteristic of Rings. Prime Fields

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.Christoph Schwarzweller - Institute of Computer Science, University of Gdańsk, PolandArtur Korniłowicz - Institute of Informatics, University of Białystok, PolandJonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565–582, 2001.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.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]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]Nathan Jacobson. Basic Algebra I. 2nd edition. Dover Publications Inc., 2009.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.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]Jarosław Kotowicz. Quotient vector spaces and functionals. Formalized Mathematics, 11 (1):59–68, 2003.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.Heinz Lüneburg. Die grundlegenden Strukturen der Algebra (in German). Oldenbourg Wisenschaftsverlag, 1999.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265–269, 2001.Michał Muzalewski. Opposite rings, modules and their morphisms. Formalized Mathematics, 3(1):57–65, 1992.Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643–648, 1991.Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.Karol Pąk. Linear map of matrices. Formalized Mathematics, 16(3):269–275, 2008. doi:10.2478/v10037-008-0032-0. [Crossref]Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.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.Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69–79, 1998.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]Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341–347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.B.L. van der Waerden. Algebra I. 4th edition. Springer, 2003.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990