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.