Finite-Dimensional Division Algebras

Платонов, В. П.; Yanchevskii, V. I.

doi:10.1007/978-3-662-03235-0_2

Cited by 15 publications

(14 citation statements)

References 61 publications

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“…Parts (1) and (3) of Theorem 2 are found in a little different context in Faddeev [3] and Platonov Yanchevskii [6]. But they considered Br(K(x)) by mod Br(K), and stated the results corresponding to (1) and (3) in an analogous way as in the case of global fields.…”

Section: (T)mentioning

confidence: 89%

The Brauer Group of a Rational Function Field over a Perfect Field

Yamasaki

1997

Journal of Number Theory

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Section: (T)mentioning

confidence: 89%

The Brauer Group of a Rational Function Field over a Perfect Field

Yamasaki

1997

Journal of Number Theory

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“…Thus in these two cases the group SK 1 (A, τ ) defined in (1) is simplified to Σ τ /Σ τ . In fact, in these settings, the reduced unitary Whitehead group is defined from the outset as SK 1 (A, τ ) = Σ τ /Σ τ (see [14], [11,Chapter 4,3.4], [8,17.B] and [5]) and so justify our definition in (1).…”

Section: R Hazratmentioning

confidence: 99%

“…This group has been studied extensively when A is a division algebra, or a graded division algebra (which is an Azumaya algebra) in literature (see [10,11] and [4]). …”

mentioning

confidence: 99%

Stability of unitary SK1 of Azumaya algebras

Hazrat

2011

Arch. Math.

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“…The following conjecture: the exponent of A is equal to its index for any division algebra A over a C 2 -field F (see for example [PY,3.4.5. ]) has the positive answer for F = F 0 ((t 1 ))((t 2 )).…”

Section: A Zheglovmentioning

confidence: 99%

Higher local skew fields

Zheglov¹

2000

Geometry &Amp; Topology Monographs

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n-dimensional local skew fields are a natural generalization of n-dimensional local fields. The latter have numerous applications to problems of algebraic geometry, both arithmetical and geometrical, as it is shown in this volume. From this viewpoint, it would be reasonable to restrict oneself to commutative fields only. Nevertheless, already in class field theory one meets non-commutative rings which are skew fields finite-dimensional over their center K . For example, K is a (commutative) local field and the skew field represents elements of the Brauer group of the field K (see also an example below). In [Pa] A.N. Parshin pointed out another class of non-commutative local fields arising in differential equations and showed that these skew fields possess many features of commutative fields. He defined a skew field of formal pseudodifferential operators in n variables and studied some of their properties. He raised a problem of classifying non-commutative local skew fields.In this section we treat the case of n = 2 and list a number of results, in particular a classification of certain types of 2-dimensional local skew fields. Basic definitionsDefinition. A skew field K is called a complete discrete valuation skew field if K is complete with respect to a discrete valuation (the residue skew field is not necessarily commutative). A field K is called an n-dimensional local skew field if there are skew fields K = K n , K n−1 , . . . , K 0 such that each K i for i > 0 is a complete discrete valuation skew field with residue skew field K i−1 . Examples.(1) Let k be a field. Formal pseudo-differential operators over k((X)) form a 2dimensional local skew field K = k((X))((∂ −1 X )), ∂ X X = X∂ X + 1. If char (k) = 0 we get an example of a skew field which is an infinite dimensional vector space over its centre.

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Finite-Dimensional Division Algebras

Cited by 15 publications

References 61 publications

The Brauer Group of a Rational Function Field over a Perfect Field

The Brauer Group of a Rational Function Field over a Perfect Field

Stability of unitary SK1 of Azumaya algebras

Higher local skew fields

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