Irreducible question

Question

Can anyone show me how to prove that $y-x^3$ is irreducible in $\mathbb{A}^2(\mathbb{C})$

For my clarity, the questions asks to decompose variety $V(xy^4-x^7y^2) \subset \mathbb{A}^2(\mathbb{C}) $ into irreducible components

Answer 1

If you mean that you want to show that $f(X, Y) = Y - X^3$ is irreducible over $\mathbb{C}[X, Y]$, you can proceed a few ways.

Using the isomorphism $\mathbb{C}[X, Y] \cong (\mathbb{C}[X])[Y]$, you can view $f$ a polynomial in $Y$ with coefficients in $\mathbb{C}[X]$. Since $f$ is linear in $Y$, it is clearly irreducible over $(\mathbb{C}[X])[Y]$, and hence over $\mathbb{C}[X, Y]$.

If you prefer, you could also use the isomorphism $\mathbb{C}[X, Y] \cong (\mathbb{C}[Y])[X]$ to view $f$ a polynomial in $X$ with coefficients in $\mathbb{C}[Y]$. Note $f$ is irreducible if and only if $-f = X^{3} - Y$ is irreducible. Applying Eisenstein's criterion to the prime ideal $\langle Y \rangle \subset \mathbb{C}[Y]$, $f$ is irreducible over $(\mathbb{C}[Y])[X]$, and hence over $\mathbb{C}[X, Y]$.