The Skolem-Noether Theorem states that any two algebra embeddings of a simple algebra into a finite-dimensional central simple algebra over the same field are related by an inner automorphism of the larger algebra, meaning conjugation by some invertible element carries one embedding onto the other. Named for Thoralf Skolem and Emmy Noether, it is a foundational uniqueness result for central simple algebras, showing their internal symmetries are as rigid as possible.
Facts
StatementGiven k-algebra homomorphisms f, g : A to B, there exists a unit b in B such that for all a in A: g(a) = b times f(a) times b to the power of negative 1. 1 Classification
Statement Form Statement Form Connections
Has Statement Form
In Branch
Source Skolem-Noether theorem, Wikipedia
Proved By
Sources
1. Skolem-Noether theorem, Wikipedia
Statement
If the dimension of B over k is finite, i.e. if B is a central simple algebra, and A is also a k-algebra, then given k-algebra homomorphisms f, g : A → B, there exists a unit b in B such that for all a in A g(a) = b · f(a) · b-1.
History section
The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme
- In Branch: Ring Theory, Lead sentence
In Group: Ring Theory and Commutative Algebra, Lead section
In ring theory, a branch of mathematics, the Skolem-Noether
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