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Division algebras. --- Gaussian sums. --- Number theory

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Ordered algebraic structures --- Division algebras. --- Algèbre à division. --- 512.55 --- Algebras, Division --- Algebraic fields --- Quasigroups --- Rings (Algebra) --- Rings and modules --- 512.55 Rings and modules --- Algèbre à division. --- Division algebras --- Algèbres à division.

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This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of noncrossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.

Mathematics. --- Associative Rings and Algebras. --- Field Theory and Polynomials. --- Algebra. --- Field theory (Physics). --- Mathématiques --- Algèbre --- Champs, Théorie des (Physique) --- Mathematics --- Algebra --- Field theory (Physics) --- Physical Sciences & Mathematics --- Valuation theory. --- Division algebras. --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Associative rings. --- Rings (Algebra). --- Mathematical analysis --- Algebraic fields --- Algebraic number theory --- Topological fields --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété