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- Jun 22, 2012

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Exercise 3(a) reads as follows:

In \(\displaystyle \mathbb{Q}[x,y] \) show the following equality of ideals:

<x + y, x - y > = <x, y>

I would appreciate help with this problem.

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My 'solution' (of which I am most unsure!!!) is as follows:

Idea generated by x + y, x - y is the ideal

\(\displaystyle h_1 ( x + y) + h_2 (x - y)\) where \(\displaystyle h_1, h_2 \in \mathbb{Q}[x,y] \)

Ideal generated by x, y is the ideal

\(\displaystyle h_3 x + h_4 y \) where \(\displaystyle h_3, h_4 \in \mathbb{Q}[x,y] \)

So

\(\displaystyle h_1(x + y) + h_2 (x - y) = h_1x + h_1y + h_2x - h_2y \)

\(\displaystyle = (h_1 + h_2)x + (h_1 - h_2)y \)

\(\displaystyle = h_3x + h_4y \)

\(\displaystyle <x,y> \)

Can someone please either correct this reasoning or confirm that is is correct/adequate.

Peter

[This is also posted on MHF]