# Base Change of Algebraic Group

I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of finite type over a field"(p. 17). Here the excerpt: Let $$G$$ be a algebraic group over a field $$k$$. Denote by $$G_{k^a}$$ the base change (or in other words coefficient extension) of $$G$$ to the algebraic closure $$k^a$$ of $$k$$.

My questions:

1. We assume that there is some point of $$G$$ lies in more than one irreducible component of $$G$$, why then the same is true for $$G_{k^a}$$?

2. If I understood it correctly then the base change step from $$G$$ to $$G_{k^a}$$ is only done in order to apply the homogeneity, since $$G$$ is only homoegeneous if the base field is algebraically closed. Is this the only reason for it?

1. Consider a scheme $$X$$ of finite type over a field $$k$$. Define the intersection of two closed subschemes of $$X$$ to be their fibered product. The formation of the intersection commutes with base change. Therefore, if two irreducible components of $$X$$ have nonempty intersection over $$k$$, they will have nonempty intersection over its algebraic closure. Of course, they may no longer then be irreducible, but obviously two of their irreducible components will have nonempty intersection.