I have a question about a remark from Liu's "Algebraic Geometry" at page 90:
Our Lemma 2.6 states following:
My QUESTION is how the author concretely deduces remark 3.1.12?
My ideas: Wlog $X_K =X$ (so $k=K$). Assume that $X_{\overline{k}}$ is reduced (or connected or irreducible) and $X$ not. The statement says that then there ixist an intermediate field extension $k=K \subset K'$ and a unique reduced (resp connected or irreducible) component $Z \subset X_{K'}$ with $Z_{\overline{k}}=X_{\overline{k}}$. Why is this a contradiction to assumption that $X_{\overline{k}}$ is reduced (or connected or irreducible)?