Ideals in Q[x, y] .... Cox, Little and O'Shea, Chapter 1 Section 4

[Math Amateur]

I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...

I am currently focused on Chapter 1, Section 4: Ideals ... ... and need help with Exercise 3(c) which reads as follows:
View attachment 5672I would be grateful if someone could help me with Exercise 3c ...

Help will be appreciated ... ...

Peter

 

[GJA]

Hi Peter,

I would be grateful if someone could help me with Exercise 3c ...

We want to show that each ideal contains the other, and, according to Problem 2, it suffices to show that the generators of one ideal are elements of the other ideal. With this in mind, try writing $x^2 - 4$ and $y^2 -1$ as elements of $\langle 2x^2 + 3y^2 -11, x^2 - y^2 -3\rangle$, which will prove $\langle x^2 - 4, y^2 - 1\rangle \subseteq \langle 2x^2 + 3y^2 -11, x^2 - y^2 -3\rangle.$ Then do the analogous thing to prove the opposite containment.

Let me know if you have any questions. Good luck!

 

[Math Amateur]

Hi Peter,
We want to show that each ideal contains the other, and, according to Problem 2, it suffices to show that the generators of one ideal are elements of the other ideal. With this in mind, try writing $x^2 - 4$ and $y^2 -1$ as elements of $\langle 2x^2 + 3y^2 -11, x^2 - y^2 -3\rangle$, which will prove $\langle x^2 - 4, y^2 - 1\rangle \subseteq \langle 2x^2 + 3y^2 -11, x^2 - y^2 -3\rangle.$ Then do the analogous thing to prove the opposite containment.

Let me know if you have any questions. Good luck!

Thanks GJA ...

That bit of help and encouragement enabled me to solve the problem ...

... thanks again ...

Peter