Re: This ideal is Maximal in Q[X,Y] ?

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 14:02:49 +0000

Eric_Smith wrote:

>
> Hello. Could you help me to solve this problem?
>
> An ideal generated by Y^2-2 and X^2+Y is maximal in Q[X,Y] where Q is
> rational.

Call I the ideal and let R = Q[X,Y]. Then in any homomorphism f from
R/I to a field K, you need f(X)^2 = 2 and f(X)^2 = -f(Y).
Take K to be the C, the complex numbers and f(X) to be sqrt(2) and
f(Y) to be i 2^{1/4}. OK the kernel of this f contains I.
Its image is the field Q(i 2^{1/4}). You need to show that the kernel
is all of I.

How to attack this? Given a polynomial G(X,Y) with rational
coefficients, prove that
G(X,Y) = a + bX + c Y + d XY + U(X,Y)(Y^2 - 2) + V(X,Y)(X^2 + Y)
for suitbale polynomials U and V etc. ... 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
  "Elegance is an algorithm"
    Iain M. Banks, _The Algebraist_

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