Re: Function is irreducible

In article <1112391622.164271.205880@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<jamesdickerson00@xxxxxxxxxxx> wrote:
>I can't get anywhere with this problem.
>
>The problem is Let F be any field and let g,h belong to F[X] be coprime
>polynomials of positive degree. Let Y be a new indeterminate and let E
>be a field of fractions for F[Y], so that F is a subset of F[Y] and
>F[Y] is a subset of E and we can view g,h in E[X]. Define f in E[X] by
>f(X)=g(X)-Yh(X). Show that f is irreducible in E[X].

Use Gauss's Lemma to show that it is irreducible in E[X]= F(Y)[X] if
and only if it is irreducible in F[Y][X]. But F[Y][X] is the same as
F[X,Y]; and it is trivially irreducible in F[X,Y] because it is linear
in Y and the coefficients are coprime in F[X].

--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

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