In the article we introduce a valuation function over a field [1]. Ring of non negative elements and its ideal of positive elements have been also defined.
@article{bwmeta1.element.doi-10_2478_v10037-012-0002-4,
author = {Grzegorz Bancerek and Hidetsune Kobayashi and Artur Korni\l owicz},
title = {Valuation Theory. Part I},
journal = {Formalized Mathematics},
volume = {20},
year = {2012},
pages = {7-14},
zbl = {1276.12005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-012-0002-4}
}
Grzegorz Bancerek; Hidetsune Kobayashi; Artur Korniłowicz. Valuation Theory. Part I. Formalized Mathematics, Tome 20 (2012) pp. 7-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-012-0002-4/
Emil Artin. Algebraic Numbers and Algebraic Functions. Gordon and Breach Science Publishers, 1994. | Zbl 1091.11001
Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.
Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Józef Białas. Properties of fields. Formalized Mathematics, 1(5):807-812, 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.
Artur Korniłowicz. Quotient rings. Formalized Mathematics, 13(4):573-576, 2005.
Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.
Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
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.
Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.