Some results on ordered fields
type: Item
instance of: scholarly article
title: Some results on ordered fields
zbMATH Open document ID: 0583.12019
author: Hiroshi Matsuyama
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: In this paper we introduce two invariants of an ordered field and show several properties concerning those invariants. In section 3 we define ''order-bases'' of an ordered field K over its subfield L, which is analogous to ''transcendency bases'', and then we show that the order type of them is determined only by \(K\) and L. Using this fact, we estimate the cardinality of the conjugacy classes of involutions in Aut(F), the automorphism group of an algebraically closed field F of characteristic 0. As is well known, the conjugacy classes of involutions in Aut(F) correspond in one-to-one manner to the isomorphism classes of subfields of F with codimension 2. Therefore, as an easy consequence, the cardinality of isomorphism classes of subfields with codimension 2 in the complex number field is greater than the continuum. In section 4 we treat a special class of subfields of K (unifinitely closed subfields), and define an equivalence relation on them. Then the set of equivalence classes admits a total ordering, whose order type, we prove in section 5, has an intimate connection with the order type of order-bases in the case \(K\) is a real closed field.
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 3933193
zbMATH Keywords: Artin-Schreier theory
zbMATH Keywords: involutions in automorphism group
zbMATH Keywords: invariants
zbMATH Keywords: ordered field
zbMATH Keywords: order-bases
zbMATH Keywords: order type
zbMATH Keywords: unifinitely closed subfields
zbMATH Keywords: real closed field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03830361
OpenAlex ID: W2009514680
DOI: 10.2969/JMSJ/03830361
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description: scientific article; zbMATH DE number 3933193