Orders and Generic Constructions of Units (de Gruyter Textbook)

Title
Copyright
Preface
Contents
1 Units in group rings: an introduction
	1.1 Constructions of units: elementary matrices and bicyclic units
	1.2 Construction of units: cyclotomic units and Bass units
	1.3 Examples: unit groups of some orders in number fields
	1.4 Examples: unit groups of some non-commutative orders
	1.5 Examples: group rings of groups of small order
	1.6 Finite rings
2 Representations of algebras
	2.1 Semisimple algebras
	2.2 Splitting fields
	2.3 Characteristic polynomial, trace and norm
	2.4 Brauer group
	2.5 Cohomology
	2.6 Crossed products
3 Wedderburn decomposition of semisimple group algebras
	3.1 Representations and characters of finite groups
	3.2 Some operations with characters
	3.3 Wedderburn components from character tables
	3.4 Wedderburn components from monomial characters
	3.5 Strongly monomial characters
	3.6 Induction theorems
	3.7 Brauer-Witt Theorem
	3.8 Examples
4 Dedekind domains, valuations and orders
	4.1 Localization and algebraic integers
	4.2 Dedekind domains
	4.3 Finitely generated modules over Dedekind domains
	4.4 Extensions of Dedekind domains
	4.5 Valuations
	4.6 Orders
	4.7 The discriminant
	4.8 Brauer group of a number field
5 Thegroupofunitsofanorder
	5.1 Lattices in real vector spaces
	5.2 Hey’s Theorem and Dirichlet’s Unit Theorem
	5.3 The group of units of an order is finitely generated
	5.4 The group of units of an order is finitely presented
	5.5 Subgroups of finite index
6 Cyclotomic integers
	6.1 Cyclotomic fields
	6.2 Cyclotomic units
7 Central units
	7.1 Thegroupofcentralunitsofanorder
	7.2 Large subgroups of central units: an algorithm
	7.3 Bass units as generators of large groups of units
8 Generic units
	8.1 Shifted cyclotomic polynomials
	8.2 The group of generic units
	8.3 A logarithm function
	8.4 A basis of generic units for a subgroup of finite index in
	8.5 Polynomials of small degree defining units
9 K-theory
	9.1 Grothendieck group
	9.2 The Whitehead group
	9.3 Stable range condition
	9.4 Whitehead group and the stable range condition
	9.5 Applications of K-theory to units
10 General linear groups of degree 2
	10.1 Number theoretical results
	10.2 Normality of in
	10.3 The factor group by
	10.4 The group E2(I) is of finite index in SL2(R)
11 Generators of unit groups of group rings
	11.1 Bass Unit Theorem
	11.2 Generalized bicyclic units and Bass units I
	11.3 Bicyclic units and Bass units
	11.4 Fixed point free groups and Frobenius complements
	11.5 Group rings of nilpotent groups
12 Exceptional simple components
	12.1 Components of index one
	12.2 Components of index two
	12.3 Generalized bicyclic units and Bass units II
	12.4 Normal closure of the trivial units
	12.5 Normal complements
	12.6 Examples: metacyclic groups
	12.7 Examples with insufficient Bass units and bicyclic units
13 Idempotents and central units in group rings
	13.1 Central subgroups and abelian-by-supersolvable groups
	13.2 Independent units and abelian-by-supersolvable groups
	13.3 Central subgroups and strongly monomial groups
	13.4 Independent units and strongly monomial groups
	13.5 Primitive idempotents and nilpotent groups
	13.6 Primitive idempotents and strongly monomial groups
	13.7 Some metacyclic groups 420
References
Index of Notation
Index