Marshall’s Quotient and the Arason–Pfister Hauptsatz for Reduced Special Groups

Kaique Matias de Andrade Roberto and Hugo Luiz Mariano ()
Additional contact information
Kaique Matias de Andrade Roberto: Centre for Logic, Epistemology and the History of Science (CLE), University of Campinas (UNICAMP), Campinas 13083-859, Brazil
Hugo Luiz Mariano: Institute of Mathematics and Statistics, University of Sao Paulo, São Paulo 05508-090, Brazil

Mathematics, 2025, vol. 13, issue 19, 1-15

Abstract: We provide a new proof of the Arason–Pfister Hauptsatz (APH) in the setting of reduced special groups, as developed by Dickmann and Miraglia. Our approach avoids the use of Boolean invariants and instead relies on a construction inspired by Marshall’s quotient, suitably adapted to the context of special groups. We establish structural properties of this quotient and show that it generalizes the Pfister quotient by a Pfister subgroup. Using this framework, we define iterated quadratic extensions of special groups and develop a theory of Arason–Pfister sequences. These tools allow us to prove that any anisotropic form φ ∈ I n ( G ) over a reduced special group G satisfies the inequality dim ( φ ) ≥ 2 n , where I n ( G ) denotes the n -th power of the fundamental ideal of the Witt ring of G . Our methods are purely algebraic and internal to the theory of special groups, contributing with novel tools to the categorical study of abstract theories of quadratic forms.

Keywords: quadratic forms; special groups; Arason–Pfister Hauptsatz; Marshall’s Quotient (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/19/3060/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/19/3060/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:19:p:3060-:d:1756119

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().