Skolem–Noether theorem

The Skolem–Noether theorem states that every automorphism of a finite-dimensional central simple algebra over a field is inner. Under suitable hypotheses, embeddings of a simple algebra into a central simple algebra are conjugate by an invertible element.

Inner automorphism: An automorphism of an algebra or group given by conjugation by one of its invertible elements. The theorem’s central claim is that every automorphism in its setting has this form.
Simple ring: A nonzero ring whose only two-sided ideals are the zero ideal and the whole ring. Simplicity is a defining hypothesis for the algebras in the theorem.
Brauer group: The group of equivalence classes of finite-dimensional central simple algebras over a field, with tensor product as its operation. Skolem–Noether helps show that choices of algebra embeddings do not alter Brauer-class constructions.
Thoralf Skolem: A Norwegian mathematician known for foundational work in logic, set theory, and algebra. Skolem’s name is attached to the theorem’s original development.
Central simple algebra: A finite-dimensional simple algebra whose center consists exactly of scalar multiples of the identity. These algebras are the theorem’s main setting.
Center of a ring: The set of ring elements that commute with every element of that ring. The theorem requires the source algebra’s center to be the base field.
Separable algebra: A finite-dimensional algebra that becomes a product of matrix algebras over an algebraic closure. The theorem supplies conjugacy results used when embedding separable algebras into central simple ones.
Emmy Noether: A German mathematician whose work transformed abstract algebra and theoretical physics. Noether’s name is attached to the theorem and its influential algebraic setting.
Centralizer: The subalgebra of elements that commute with every element of a specified subset. Centralizers help describe the conjugating elements used for embeddings.
Field extension: A field containing a specified subfield. The base field and its extensions set the scalars for the algebras and embeddings.

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Division ringRelated: Finite-dimensional division algebras over their centers are central simple algebras.