# Zariski Topology and Regular Functions on Algebraic Varieties in Affine Space

i) Show that the Zariski topology on $Spec(k[x,y])$ is NOT the product of the Zariski topologies on $Spec(k[x])\times Spec(k[y])$ (unlike the metric topology, in wich the product of the topologies is equivalent to the topology of the product).

ii) Show that the ring of regular functions $A(P)$ of the parabola $P$ defined by $y=x^2$ in $\mathbb{A}^2_k:=Spec[x,y]$ is isomorph to ring of polynomials in one variable over $k$.

iii) Show that the ring of regular functions $A(P)$ of the plave curve $C$ defined by $x\cdot y=2$ in $\mathbb{A}^2_k$ is NOT isomorph to ring of polynomials in one variable over k.

iv) Let $C$ be a curve in $\mathbb{A}^3_k$ given by the parameterization $\{(t,t^2,t^3)|t\in k\}$.

a) Find generators of the ideal $I(C)$.

b) Show that $A(C)$ is isomorph to ring of polynomials in one indeterminate.

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