section_path: Contributions to mathematics and physics/Third epoch (1927–1935)/Noncommutative algebra
section_name: Noncommutative algebra
paragraph_text: Noncommutative algebra
  
  Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.
  
  A paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras, which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem"):
  
  every finite dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.
  
  These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field k into a finite-dimensional central simple algebra over k are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.