Algebraic Group with two components over algebraically closed field

Question

I want to find an algebraic group with two components over, say, $\mathbb{C}$ for simplicity. This more or less means that we want an index 2 subgroup. Wikipedia says that the character group $X^*(\mathbb{G}_m) \approx \mathbb{Z}$, so this should work since we have the subgroup $2\mathbb{Z}$. However, I do not understand why this is correct (I think the isomorphism is true since all of the maps are power maps of some degree) - in particular, a variety should be the zero locus of a finite set of (finite degree) polynomials, so I do not see how $\mathbb{Z}$, or even $\text{Spec}\mathbb{Z}$, could satisfy this.