I wanted to show the elementary fact mentionned in the title.
Context: Let $\mathbb{k}$ be an algebrically closed field of caracteristic 0. (I know that I don't need that much hypothesis, but these are the general hypothesis I'm working on). Then the multiplicative group $\mathbb{G}_m \simeq \mathbb{k} \backslash \{0\}$ is a connected algebraic group.
I Know for sure that $\mathbb{G}_m$ is an affine standard set (i.e. $\mathbb{G}_m \simeq D(P) = \mathbb{k} \backslash V(P)$) where $P$ is the polynomial $P(T) = T$. I have the intuition that the proof of the connectedness lies in here, but I don't know how to explicit it. Maybe a general fact that an affine standard open set is irreductible?
Could you enlight me please? Thanks in advance.
K. Y.