Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the ring of polynomials $k[x,y,z]$, $k$ a field.

Why is it clear that $x \not \in (x^{2},xz,z^{2},xy-z^{2})$ ?

I mean, how do we show this rigorously? so assume it belongs, then:

$x = f(x,y,z)x^{2} + g(x,y,z)xz + h(x,y,z)z^{2} + t(x,y,z)(xy-z^{2})$ where $f,g,h,t$ are elements of $k[x,y,z]$, yes?

Now from here I'm not sure how to proceed and derive a contradiction. I tried setting $x=0, y=0$ and $z$ fixed so:

$0=h(0,0,z)z^{2} + t(0,0,z)z^{2}$ and thus $0=z^{2}p(z)$ where $p(z)=h(0,0,z)+t(0,0,z)$.

Now this can only happen when $p(z)=0$ but how we know this cannot happen?

What is an elegant way (or quick one) to see that $x$ does not belong to such ideal?

share|improve this question
2  
Consider the degrees of the monomials in the summands. None of them contains any linear (degree 1) terms at all, so... –  Alon Amit Apr 28 '11 at 19:59
add comment

1 Answer

Hint: Look at the degrees of the polynomials on both sides of the equation you get after assuming that $x$ is in the ideal.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.