109,096 research outputs found
Ordered Rings and Fields
We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].Schwarzweller Christoph - Institute of Informatics, University of Gdansk, Gdansk, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363-371, 2016.Nathan Jacobson. Lecture Notes in Abstract Algebra, III. Theory of Fields and Galois Theory. Springer-Verlag, 1964.Manfred Knebusch and Claus Scheiderer. Einf¨uhrung in die reelle Algebra. Vieweg-Verlag, 1989.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984.Knut Radbruch. Geordnete K¨orper. Lecture Notes, University of Kaiserslautern, Germany, 1991.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990
The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields
For , 4, and 5, we prove that, when -number fields of degree
are ordered by their absolute discriminants, the lattice shapes of the rings of
integers in these fields become equidistributed in the space of lattices.Comment: 12 page
Definable valuations on ordered fields
We study the definability of convex valuations on ordered fields, with a
particular focus on the distinguished subclass of henselian valuations. In the
setting of ordered fields, one can consider definability both in the language
of rings and in the richer language of ordered rings
. We analyse and compare definability in both
languages and show the following contrary results: while there are convex
valuations that are definable in the language but
not in the language , any
-definable henselian valuation is already
-definable. To prove the latter, we show that the
value group and the ordered residue field of an ordered henselian valued field
are stably embedded (as an ordered abelian group, respectively as an ordered
field). Moreover, we show that in almost real closed fields any
-definable valuation is henselian.Comment: 17 page
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