×

Defining transcendentals in function fields. (English) Zbl 1015.03041

There are two notions of “relative algebraic closure” of a field \(K\) in a field extension \(F/K\): the field-theoretic relative algebraic closure \[ \overline{K} \cap F = \{x\in F \mid f(x)=0\text{ for some } f\in K[T] \} \] and the model-theoretic relative algebraic closure \[ \text{acl}_{F}(K):=\{ x\in F \mid x\in A \text{ for some finite }K\text{-definable }A\subseteq F \} \] (where “\(K\)-definable” means definable by a first-order formula in the language of fields with parameters from \(K\)).
Clearly, \(\overline{K}\cap F\subseteq \text{acl}_F(K)\); the present paper provides examples of field extensions \(F/K\) where this is a proper inclusion (i.e., where the two notions of relative algebraic closure of \(K\) in \(F\) do not coincide, or, equivalently, where \(F\) contains \(K\)-definable transcendentals over \(K\)).

MSC:

03C60 Model-theoretic algebra
12L12 Model theory of fields
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Algebraic function fields and codes (1993)
[2] Aus einem noch nicht veröffentlichten Briefe an Herrn P. Schwarz II pp 235–
[3] Journal für die Reine und Angewandte Mathematik 179 pp 5– (1938)
[4] Mathematische Annalen 41 pp 403– (1893)
[5] Algebraic Curves (1969)
[6] Sur la théorie élémentaire des corps de fonctions 51 pp 948– (1986)
[7] DOI: 10.1007/978-3-540-68521-0_9
[8] Introduction to the theory of algebraic functions of one variable (1951)
[9] Journal für die Reine und Angewandte Mathematik 427 pp 107– (1992)
[10] DOI: 10.1090/S0002-9939-1992-1095220-5
[11] Indécidabilité des corps de courbe réelle 59 pp 87– (1994)
[12] Définissabilité dans les corps de fonctions p-adiques 56 pp 783– (1991)
[13] Quadratic and hermitian forms pp 270– (1985)
[14] DOI: 10.1090/S0002-9947-1980-0583852-6
[15] DOI: 10.1090/S0002-9947-1955-0070211-3
[16] Archiv der Mathematik III pp 343– (1952)
[17] DOI: 10.1002/malq.19640101803 · Zbl 0221.02034
[18] Definability and decision problems in arithmetic 14 pp 98– (1949) · Zbl 0034.00801
[19] DOI: 10.2307/2118581 · Zbl 0862.12003
[20] DOI: 10.1007/BF02402192 · JFM 17.0279.01
[21] Soviet Mathematics Doklady 11 pp 354– (1970)
[22] DOI: 10.2969/jmsj/00310137 · Zbl 0044.26901
[23] The arithmetic of elliptic curves (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.