Let E / k be a field extension. If A and B are central simple k algebras with A ∼B, prove that E

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Let E / k be a field extension. If A and B are central...

Key Concepts

Morita Theory

Morita theory provides insights into when two algebras can be considered 'the same' from the viewpoint of module theory, meaning their categories of modules are equivalent. In the context of central simple algebras, Morita theory explains how two seemingly different algebras might still represent the same class under Brauer equivalence, since each is Morita equivalent to the division algebra over which they are matrix algebras. This theoretical framework underpins many results regarding the preservation of equivalence under operations like tensor product.

Field Extensions and Scalar Extension

A field extension, in this context, involves taking a base field and extending it to a larger field. Scalar extension in algebra is the process of 'extending scalars', meaning one takes an algebra over a base field and forms a new algebra over an extended field by tensoring with the field extension. This process is crucial because it allows the behavior of algebraic structures to be studied over different fields, often simplifying the investigation of properties like simplicity and centrality.

Tensor Products of Algebras

The tensor product operation is used to combine algebras over a common field in such a way that the resulting structure preserves many essential properties of the factors. When an algebra over a field is tensored with a field extension, the resulting algebra retains key structural characteristics (such as being central simple) while now being defined over the larger field. This operation is fundamental in transferring problems and properties between different field contexts.

Central Simple Algebras

A central simple algebra is a finite-dimensional algebra over a field that has no non-trivial two-sided ideals and whose center is exactly the base field. These algebras play a central role in algebra and are key objects in understanding division algebras and structures like matrix algebras. Their simplicity means they have a rigid structure, making them building blocks in the theory of algebras.

Brauer Equivalence

Brauer equivalence relates two central simple algebras that, although possibly not isomorphic as algebras, share the same 'division algebra component' when described through the Artin-Wedderburn theorem. That is, each can be realized as a matrix algebra over some division algebra, and when these division algebras are isomorphic, the two algebras are said to be Brauer equivalent. This equivalence is the basis of the Brauer group, an important invariant in the study of field extensions and division algebras.

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