Witt rings of infinite algebraic extensions of global fields
Abstract
In this paper we discuss the problem to carry over the well-known Minkowski-Hasse local-global principle to the context of an infinite algebraic extension of the rationals or the rational function fields 𝔽q(x) over finite fields. Applying this result we give a new proof of the elementary type conjecture for Witt rings of infinite algebraic extensions of global fields. This generalizes a result of I. Efrat [Ef] who proved, using Galois cohomology methods, a similar fact for algebraic extensions of the rationals.
Keywords
infinite extensions of global fields; local-global principle; Witt ring
References
2. M. Kula, Fields with prescribed quadratic form schemes, Math. Z., 167 (1979), 201-212.
3. K. Kuratowski, A.M. Mostowski, Set Theory with an Introduction to Descriptive Set Theory, PWN - Polish Scientific Publishers, Warsaw & North-Holland Publishing Company, Amsterdam-New York-Oxford (1976).
4. T.Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading (Mass.) (1980).
5. M. Marshall, Abstract Witt Rings, Queen's Papers in Pure and Applied Mathematics - No. 57, Queen's University, Kingston (1980).
6. J. Neukirch, Class field theory, Grundlehren der mathematischen Wissenschaften, 280, Springer-Verlag, Berlin-New York (1986).
7. W. Scharlau, Quadratic and Hermitian Forms, Springer Verlag, Berlin-Heidelberg-New York (1985).
8. W. Scharlau, private communication.
9. J.L. Yucas, Quadratic forms and radicals of fields, Acta Arithmetica, 39 (1981), 313-322.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
