Explore: Ordered Fields

Artin–Schreier theory of ordered fields and formally real fields. There are two equivalent common definitions of an ordered field. The definition of total

that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This

Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements

known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. Many other fields, such as fields of rational

initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are totally ordered by definition. They include

decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005). Salma Kuhlmann, Ordered Exponential Fields, Fields Institute Monographs, 12, (2000)

Levi-Civita field the hyperreal number fields the superreal number fields the field of surreal numbers (this is a proper class, not a set) If F is an ordered field

reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting

rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order

involved. More generally, this applies for an ordered field. For more information, see § Ordered fields. The property for the additive inverse states