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[maitranh]

Please help!!!!!!!!!!!!! thanks
Let F be a finite-dimensional extension of K, and let f(x) be a polynomial in F[x]. Prove there exists a polynomial g(x) in F[x] such that f(x)g(x) is in K[x].

[canh_dieu]

The question is more than a month old but let's just post the answer here with a hope that the owner of the question will visit us back one day

If http://dientuvietnam...n/mimetex.cgi?F is a finite extension of http://dientuvietnam...n/mimetex.cgi?K then every http://dientuvietnam...imetex.cgi?(f(x))^m+c_1(x)(f(x))^{m-1}+\cdots+c_{m-1}(x)f(x)+c_m(x)=0
which obviously implies that http://dientuvietnam.net/cgi-bin/mimetex.cgi?b_1,b_2,\ldots,b_m be a set of generators for http://dientuvietnam...n/mimetex.cgi?F over http://dientuvietnam.../mimetex.cgi?K. We may assume that http://dientuvietnam...etex.cgi?b_1=1. It is clear that these elements also generate F[x] as a module over K[x].

For each http://dientuvietnam...n/mimetex.cgi?i
http://dientuvietnam.net/cgi-bin/mimetex.cgi?[f(x)I-(a_{ij}(x))][b]=0
as metrices.

Multiply both sides by the adjoint matrix we get
http://dientuvietnam.net/cgi-bin/mimetex.cgi?\det[f(x)I-(a_{ij}(x))][b]=0.
Since http://dientuvietnam.net/cgi-bin/mimetex.cgi?K not neccesarily finite, because we may replace http://dientuvietnam.net/cgi-bin/mimetex.cgi?F with the subfield generated over http://dientuvietnam.net/cgi-bin/mimetex.cgi?K by coeffcients of http://dientuvietnam.net/cgi-bin/mimetex.cgi?f(x).