Re: Can anyone prove this lemma?

rupertmccallum_at_yahoo.com
Date: 12/29/04 

Date: 28 Dec 2004 17:29:07 -0800

Bill Dubuque wrote:
> rupertmccallum@yahoo.com wrote:
> >
> > Your first result, if a polynomial f(X) has coefficients in A and
> > f(u)=0 then f(X)/(X-u) has coefficients in A, follows from the
> > Magidin-McKinnon theorem that every polynomial with algebraic
integer
> > coefficients factorizes into linear polynomials with algebraic
integer
> > coefficients. I would be interested in how to prove this in an
> > elementary way.
> n-1 n-2 n-3
> Write f = (X-u)(a X + b X + c X +...), a,b,c...in K, u in A
>
> Comparing the leading terms shows a = f_n in A.
>
> Now subtract (X-u) a X^n from both sides and by
>
> induction deduce remaining coefs b,c... are in A. QED
>
> --Bill Dubuque

Who says u is in A? That would only be if f is monic.

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