In this chapter we will concentrate on lattice-ordered fields. Since more is known about totally ordered fields than about ℓ-fields in general most of this chapter will be concerned with totally ordered fields. Examples of ℓ-fields come from power series ℓ-rings and from constructing lattice orders on the reals and other similar totally ordered fields. We will develop the algebraic properties of totally ordered fields, including the existence and uniqueness of its real closure, which culminates in a description of those fields whose algebraic closure is a finite extension. In order to show that some commutative ℓ-domains can be embedded in power series ℓ-fields with real coefficients and with exponents in the associated value po-group—the Hahn Embedding Theorem—we will need to develop enough valuation theory to first carry out the embedding for totally ordered fields. We will also see that a totally ordered division ring can be enlarged to one whose center contains the reals.
KeywordsPartial Order Topological Vector Space Formal Power Series Total Order Division Ring
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