Central simple algebra

Summary

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra K[X,\partial_X] is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence

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Minimum euclidien des ordres maximaux dans les algèbres centrales à division

Jérôme Chaubert

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.

EPFL2007

Euclidean minima and central division algebras

Eva Bayer Fluckiger, Jérôme Chaubert

The notion of Euclidean minimum of a number field is a classical one. In this paper we generalize it to central division algebras and establish some general results in this new context.

2009

Descent properties of hermitian Witt groups in inseparable extensions

Eva Bayer Fluckiger, Daniel Arnold Moldovan

Let k be a field of characteristic ≠2, A be a central simple algebra with involution σ over k and W(A,σ) be the associated Witt group of hermitian forms. We prove that for all purely inseparable extensions L of k, the canonical map rL/k:W(A,σ)⟶W(AL,σL) is an isomorphism.

Springer Verlag2011