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.

Let $Y=\{(a,b,c) \in \mathbb{A}^{3}: a^{2}-a^{2}b^{2}+c^{3}=0\}$. How can we parametrize $Y$ so that we can find the irreducible components of $Y$?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

I'll change the notation and say we are studying $Y=V(x^2-x^2y^2+z^3)\subset \mathbb A^3_k$ .

The polynomial $f(x,y,z)=x^2-x^2y^2+z^3=z^3+x^2(1-y)(1+y)$ is irreducible in $k[x,y,z]=k[x,y][z]$ by Eisenstein's criterion applied to the prime $1-y\in k[x,y]$

Hence, for any field $k$, the closed subset $Y\subset \mathbb A^3_k$ is irreducible .

share|improve this answer

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.