Let $J=\langle x,y\rangle$ and $I=J^3=J\cdot J\cdot J$ in $\mathbb{Q}[x,y]$.
1.1 The set of all monomials in $x, y$ is a basis of $\mathbb{Q}[x,y]$ as $\mathbb{Q}$-vector space. In particular, $x^2$ and $x^3$ are linearly independent in the $\mathbb{Q}$-vector space $\mathbb{Q}[x,y]$. Explain why $x^2 + I$ and $x^3 + I$ are linearly dependent in the $\mathbb{Q}$-vector space $\mathbb{Q}[x,y]/I$.
1.2 Find $\sqrt{I}$ and $\dim_{\mathbb{Q}}(\mathbb{Q}[x,y]/\sqrt{I})$ and $V(\sqrt{I})\subset \mathbb{Q^2}$ by hand.
Concerning 1.2, I was thinking:
$I = J \cdot J \cdot J = J^3$ $\Rightarrow$ $\sqrt{I} = J$, since $J^3 = I$
$\sqrt{I} = \langle x,y\rangle$, $V(\sqrt{I}) = \lbrace(0,0)\rbrace$
$\mathbb{Q}[x,y]/\sqrt{I} = \lbrace 1 \rbrace$
$\Rightarrow$ $\dim_{\mathbb{Q}}(\mathbb{Q}[x,y]/\sqrt{I}) = 1$
Does that make sense?
Concerning 1.1, I don't really have a clue, so any help or explanations would be appreciated.