Separable polynomials over finite dimensional algebras
作者:
David R. Finston,
期刊:
Communications in Algebra
(Taylor Available online 1985)
卷期:
Volume 13,
issue 7
页码: 1597-1626
ISSN:0092-7872
年代: 1985
DOI:10.1080/00927878508823241
出版商: Gordon and Breach Science Publishers Ltd.
数据来源: Taylor
摘要:
In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality ndor the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly nddistinct zeros in [Ktilde] ⨷kA [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ⨷kA contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.
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