proving an algebraic set is irreducible

Question

$X=\{{t,t^2}|t\in \mathbb{C}\} \subset \mathbb{C^2}$. Show that $X$ is an irreducible set.

Trying to put $X=A \cup B$ where $A$ and $B$ are algebraic sets. I have to show either $X=A$ or $X=B$.

Thanks in advance for your help.

Answer 1

Your $X$ is the image of the morphism $\mathbb A^1\to\mathbb A^2$ sending $t\mapsto (t,t^2)$. This corresponds to the $\mathbb C$-algebra map $\phi:\mathbb C[x,y]\to \mathbb C[t]$ sending $x\mapsto t,\,y\mapsto t^2$. We have: $$X=V(\ker\phi)=V(y-x^2).$$ This algebraic set is irreducible, since $(y-x^2)\subset \mathbb C[x,y]$ is a prime ideal.

Answer 2

Here's a short proof: since $X= \{ (t,t^2) \}$ for $t \in \mathbb C$, $X$ contains the torus $\mathbb C ^\ast$ as a dense open subset. The torus is irreducible, hence $X$ must be as well.