Algebraic Varieties and Ideals

In Chapter 1, Section 4 of Cox et al "Ideals, Varieties and Algorithms, Exercise 3(c) reads as follows:

Prove the following equality of ideals in $\displaystyle \mathbb{Q}[x,y] $:

$\displaystyle < 2x^2 + 3y^2 -11, x^2 - y^2 - 3> \ = \ <x^2 - 4, y^2 - 1> $

Any help with this problem would be appreciated.

Peter

Re: Algebraic Varieties and Ideals

I will give you one half:

$\displaystyle x^2 - 4 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) + \dfrac{3}{5}(x^2 - y^2 - 3)$

$\displaystyle y^2 - 1 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) - \dfrac{2}{5}(x^2 - y^2 - 3)$

Your turn.

Re: Algebraic Varieties and Ideals

Quote:

Originally Posted by

**Deveno** I will give you one half:

$\displaystyle x^2 - 4 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) + \dfrac{3}{5}(x^2 - y^2 - 3)$

$\displaystyle y^2 - 1 = \dfrac{1}{5}(2x^2 + 3y^2 - 11) - \dfrac{2}{5}(x^2 - y^2 - 3)$

Your turn.

Thanks for the guidance and help Deveno

OK so we want to show that in $\displaystyle \mathbb{Q}[x,y] $ we have $\displaystyle <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ = \ <x^2 - 4, y^2 - 1> $

**First show that $\displaystyle <x^2 - 4, y^2 - 1> \ \subseteq \ <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> $**

From Deveno's equations above it follows that $\displaystyle x^2 - 4, y^2 - 1 \in \ <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> $

Thus, $\displaystyle <x^2 - 4, y^2 - 1> \ \subseteq \ <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> $ ... ... ... (1)

**Now show $\displaystyle <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ \subseteq \ <x^2 - 4, y^2 - 1> $**

We can write $\displaystyle 2x^2 + 3y^2 - 11 \ = \ 2(x^2 - 4) + 3(y^2 - 1) $

and $\displaystyle x^2 - y^2 - 3 \ = \ (x^2 - 4) - (y^2 - 1) $

Thus $\displaystyle 2x^2 + 3y^2 - 11, x^2 - y^2 - 3 \in <x^2 - 4, y^2 - 1> $

Thus $\displaystyle <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ \subseteq \ <x^2 - 4, y^2 - 1> $ ... ... ... (2)

Equations (1) (2) $\displaystyle \ \Longrightarrow \ \mathbb{Q}[x,y] $ we have $\displaystyle <2x^2 + 3y^2 - 11, x^2 - y^2 - 3> \ = \ <x^2 - 4, y^2 - 1> $

Re: Algebraic Varieties and Ideals

Note this shows the generators of ideals in Q[x,y] aren't even unique up to units. Also, this is just high-school algebra in fancier clothes.