Hereditarily non-pythagorean fields

David Grimm; David B. Leep

doi:10.1016/j.jalgebra.2021.01.013

Hereditarily non-pythagorean fields

David Grimm, David B. Leep

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove for a large class of fields F that every proper finite extension of F_pyth, the pythagorean closure of F, is not a pythagorean field. This class of fields contains number fields and fields F that are finitely generated of transcendence degree at least one over some subfield of F.

Original language	English
Pages (from-to)	434-452
Number of pages	19
Journal	Journal of Algebra
Volume	574
DOIs	https://doi.org/10.1016/j.jalgebra.2021.01.013
State	Published - May 15 2021

Bibliographical note

Funding Information:
We thank Karim Becher, Eberhard Becker, and Parul Gupta for useful conversations on these topics. Financial support by CONICYT (proyecto FONDECYT 11150956 ) and by the Universidad de Santiago de Chile (USACH) (proyecto DICYT, Codigo 041933G ) are gratefully acknowledged.

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Function field
Hereditarily pythagorean
Number field
Pythagorean closure
Pythagorean field
Quadratic closure
Quadratically closed field
Sums of squares
Valuations

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2021.01.013

Cite this

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title = "Hereditarily non-pythagorean fields",

abstract = "We prove for a large class of fields F that every proper finite extension of Fpyth, the pythagorean closure of F, is not a pythagorean field. This class of fields contains number fields and fields F that are finitely generated of transcendence degree at least one over some subfield of F.",

keywords = "Function field, Hereditarily pythagorean, Number field, Pythagorean closure, Pythagorean field, Quadratic closure, Quadratically closed field, Sums of squares, Valuations",

author = "David Grimm and Leep, {David B.}",

note = "Funding Information: We thank Karim Becher, Eberhard Becker, and Parul Gupta for useful conversations on these topics. Financial support by CONICYT (proyecto FONDECYT 11150956 ) and by the Universidad de Santiago de Chile (USACH) (proyecto DICYT, Codigo 041933G ) are gratefully acknowledged. Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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doi = "10.1016/j.jalgebra.2021.01.013",

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AU - Grimm, David

AU - Leep, David B.

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PY - 2021/5/15

Y1 - 2021/5/15

N2 - We prove for a large class of fields F that every proper finite extension of Fpyth, the pythagorean closure of F, is not a pythagorean field. This class of fields contains number fields and fields F that are finitely generated of transcendence degree at least one over some subfield of F.

AB - We prove for a large class of fields F that every proper finite extension of Fpyth, the pythagorean closure of F, is not a pythagorean field. This class of fields contains number fields and fields F that are finitely generated of transcendence degree at least one over some subfield of F.

KW - Function field

KW - Hereditarily pythagorean

KW - Number field

KW - Pythagorean closure

KW - Pythagorean field

KW - Quadratic closure

KW - Quadratically closed field

KW - Sums of squares

KW - Valuations

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DO - 10.1016/j.jalgebra.2021.01.013

M3 - Article

AN - SCOPUS:85100680835

VL - 574

SP - 434

EP - 452

ER -

Hereditarily non-pythagorean fields

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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