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Tuesday, May 14, 2013

Polynomial Ideals and Primality (9.1.13)

Dummit and Foote Abstract Algebra, section 9.1, exercise 13:

MathJax TeX Test Page Letting F be a field, prove F[x,y]/(y2x)F[x,y]/(y2x2).

Proof: We show something a bit stronger, that F[x,y]/(ynx)F[x,y]/(x2y2) for any n, with F only an integral domain.

First we show yx,y+x(y2x2)=(yx)(y+x) so that there are zero divisors in F[x,y]/(y2x2). Assuming the contrary we have for f some polynomial yx=(yx)(y+x)f(y+x)f=1, an impossibility. Similarly for y+x(y2x2).

Now we show F[x,y]/(ynx)=F[y]/(ynx)F[y] is an integral domain, which will complete the proof. To see the first equality, notice that is clear and any polynomial of F[x,y]/(ynx) can be represented by the image of a polynomial devoid of variables in x since ¯yn=¯x. Letting φ be the natural surjective homomorphism of F[y] onto F[y]/(ynx), we can observe that if ker φ0 then 0(ynx)fF[y] for some fF[x,y], an impossibility by (ynx)f=ynfxf and observing the result of the xf term by choosing the result of the multiplication monomial with maximal degree of x in f, which cannot be affected by any terms in xf (else the f monomials' y and x degrees are the same and can be rewritten as one) nor in ynf. 

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