Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.
@article{bwmeta1.element.doi-10_2478_forma-2013-0013,
author = {Yuichi Futa and Hiroyuki Okazaki and Daichi Mizushima and Yasunari Shidama},
title = {Gaussian Integers},
journal = {Formalized Mathematics},
volume = {21},
year = {2013},
pages = {115-125},
zbl = {1298.11009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0013}
}
Yuichi Futa; Hiroyuki Okazaki; Daichi Mizushima; Yasunari Shidama. Gaussian Integers. Formalized Mathematics, Tome 21 (2013) pp. 115-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0013/
[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
[2] Grzegorz Bancerek. Konig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.
[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
[4] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
[5] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
[6] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
[7] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
[8] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[9] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
[10] Czesław Bylinski. 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] | Zbl 1276.11090
[13] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.[Crossref] | Zbl 1276.94012
[14] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
[15] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
[16] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
[17] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.
[18] 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.
[19] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.
[20] Christoph Schwarzweller. The field of quotients over an integral domain. Formalized Mathematics, 7(1):69-79, 1998.
[21] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
[22] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
[23] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
[24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
[25] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[27] Andr´e Weil. Number Theory for Beginners. Springer-Verlag, 1979.
[28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.