Non-Commutative Polynomials and Cyclic Algebras
Nathan S. Jacobson · Annals of Mathematics · 1934
where a is an element of a' such that the correspondence a into a(l) generates the Galois group of a' over a. We consider a'(II) as a residue algebra of the polynomial H in the domain a'[x] of all polynomials in x whose coefficients C a' and for which multiplication is defined by (1). The main purpose of the present paper is to show that the problem of determining the division algebra part of a'(H) in the Wedderburn decomposition may be reduced to the problem of finding a single irreducible factor of II in a' [x]. In ?1 we discuss the factorization theory in a' [x]. In ?2 and ?3 we determine the relation between a'(H) and an algebra associated with a factor of H. Finally, in ?4 we derive a condition in order that H have an irreducible factor of degree t and from this condition the usual norm conditions for cyclic algebras follow as simple corollaries. The methods used are applicable to the algebras defined by (1) and (2) where a' is any division algebra of finite order over 0. We hope to consider this case in a later paper. In the course of the preparation of this paper, I had the privilege of discussing its details with Professor Wedderburn. I am very grateful to him for the stimulus of these discussions.