logo CRAS
Comptes Rendus. Mathématique
Mathematical logic
Quantifier elimination for quasi-real closed fields
Mickaël Matusinski1; Simon Müller2 
1 Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
2 Universität Konstanz, 78467 Konstanz, Germany
Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 291-295.

We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.

Nous prouvons l’élimination des quantificateurs pour la théorie des corps quasi-réels clos munis d’une valuation compatible. Cela reprend et unifie les mêmes résultats connus pour les corps algébriquement clos et les corps réels clos.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.169
Classification: 03C10,  03C64,  12J10,  12J15,  12L12
Mickaël Matusinski 1; Simon Müller 2

1 Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
2 Universität Konstanz, 78467 Konstanz, Germany 
@article{CRMATH_2021__359_3_291_0,
     author = {Micka\"el Matusinski and Simon M\"uller},
     title = {Quantifier elimination for quasi-real closed fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {291--295},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.169},
     language = {en},
}
TY  - JOUR
TI  - Quantifier elimination for quasi-real closed fields
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 291
EP  - 295
VL  - 359
IS  - 3
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.169
DO  - 10.5802/crmath.169
LA  - en
ID  - CRMATH_2021__359_3_291_0
ER  - 
%0 Journal Article
%T Quantifier elimination for quasi-real closed fields
%J Comptes Rendus. Mathématique
%D 2021
%P 291-295
%V 359
%N 3
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.169
%R 10.5802/crmath.169
%G en
%F CRMATH_2021__359_3_291_0
Mickaël Matusinski; Simon Müller. Quantifier elimination for quasi-real closed fields. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 291-295. doi : 10.5802/crmath.169. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.169/

[1] Emil Artin; Otto Schreier Algebraische Konstruktion reeller Körper, Abh. Math. Semin. Univ. Hamb., Volume 5 (1927) no. 1, pp. 85-99 | Article | Zbl: 52.0120.05

[2] Reinhold Baer Über nicht-archimedisch geordnete Körper. (Beiträge zur Algebra 5.)., Volume 8, 1927, pp. 3-13 | Zbl: 53.0118.01

[3] Gregory Cherlin; Max A. Dickmann Real closed rings II, Ann. Pure Appl. Logic, Volume 25 (1983) no. 3, pp. 213-231 | Article | MR: 730855 | Zbl: 0538.03028

[4] Charles N. Delzell; Alexander Prestel Mathematical logic and model theory. A brief introduction, Universitext, Springer, 2011, x+193 pages | Zbl: 1241.03001

[5] Ido Efrat Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs, 124, American Mathematical Society, 2006, xiv+288 pages | Article | MR: 2215492 | Zbl: 1103.12002

[6] Antonio J. Engler; Alexander Prestel Valued fields, Springer Monographs in Mathematics, Springer, 2005, x+205 pages | Zbl: 1128.12009

[7] Syed M. Fakhruddin Quasi-ordered fields, J. Pure Appl. Algebra, Volume 45 (1987) no. 3, pp. 207-210 | Article | MR: 890020 | Zbl: 0629.12022

[8] Wolfgang Krull Allgemeine Bewertungstheorie, J. Reine Angew. Math., Volume 167 (1932), pp. 160-196 | Article | MR: 1581334 | Zbl: 0004.09802

[9] Salma Kuhlmann; Mickaël Matusinski; Françoise Point The Valuation Difference Rank of a Quasi-Ordered Difference Field, Groups, Modules, and Model Theory – Surveys and Recent Developments : In Memory of Rüdiger Göbel (Manfred Droste; László Fuchs; Brendan Goldsmith; Lutz Strüngmann, eds.), Springer, 2017, pp. 399-414 | Article | Zbl: 1436.03202

[10] Salma Kuhlmann; Simon Müller Compatibility of Quasi-Orderings and Valuations: A Baer–Krull Theorem for Quasi-Ordered Rings, Order, Volume 36 (2019) no. 2, pp. 249-269 | Article | MR: 3983478 | Zbl: 07083960

[11] Abraham Robinson Complete theories, Studies in Logic and the Foundations of Mathematics, North-Holland, 1956 | Zbl: 0070.02701

[12] Joseph Shipman Improving the fundamental theorem of algebra, Math. Intell., Volume 29 (2007) no. 4, pp. 9-14 | Article | MR: 2361620 | Zbl: 1158.00301

[13] Oscar Zariski; Pierre Samuel Commutative algebra. Vol. II, Graduate Texts in Mathematics, 29, Springer, 1975, x+414 pages (Reprint of the 1960 edition) | MR: 389876

Cited by Sources: