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doctoral thesis

Minimum euclidien des ordres maximaux dans les algèbres centrales à division

Chaubert, Jérôme  
2007

This work concerns the study of Euclidean minima of maximal orders in central simple algebras. In the first part, we define the concept of ideal lattice in the non-commutative case. Let A be a semi-simple algebra over Q. An ideal lattice over A is a triple (I, α, τ) where I is an ideal of A, α is a unit in AR = A ⊗Q R fixed by τ and τ is a positive involution on AR, in other words, trAR/R(xαxτ) > 0 for all x ∈ IR. We study ideal lattices, especially their Hermite invariant, to link the concepts of Euclidean minimum of a maximal order and ideal lattices of the form (Λ, α, τ). Namely, we show that we can bound the Euclidean minimum of Λ by the Hermite invariant of the ideal lattices of Λ. This upper bound allows us to calculate the Euclidean minima of some infinite families of maximal orders, when A is a quaternion skew field over Q. In the second part, we look for other ways to find upper and lower bounds of the euclidean minimum of a maximal order in a quaternion skew field. This research leads us to an exhaustive list of euclidean quaternion skew fields over Q, and their euclidean minima. The quadratic case has also been studied but remains partially unsolved. At the end, we give bounds for Euclidean minima of some maximal orders of quaternion skew fields over cyclotomic fields.

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Type
doctoral thesis
DOI
10.5075/epfl-thesis-3717
Author(s)
Chaubert, Jérôme  
Advisors
Bayer Fluckiger, Eva  
Jury

Donna Testerman, Jean-Claude Belfiore, Jean-Paul Cerri

Date Issued

2007

Publisher

EPFL

Publisher place

Lausanne

Public defense year

2007-01-11

Thesis number

3717

Total of pages

204

Subjects

Euclidean minimum

•

central simple algebra

•

division algebra

•

quaternions

•

quaternion skew fields

•

maximal order

•

ideal lattice

•

minimum euclidien

•

algèbre centrale simple

•

algèbre à division

•

quaternions

•

corps de quaternions

•

ordre maximal

•

réseau idéal

•

anneau euclidien

EPFL units
CSAG  
Faculty
SB  
Doctoral School
EDMA  
Available on Infoscience
November 24, 2006
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/235967
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