I'm currently learning Algebraic K Theory from Milnor's "An Introduction to Algebraic K Theory" and am having trouble understanding his proof of Rim's theorem using a "Mayer-Vietoris" exact sequence. Here is an image of the section:
By "Hypothesis 1 and 2" Milnor is referring to a Milnor square:
(Sorry for the poor images). Firstly, I'm unsure how hypothesis $1$ is satisfied. In Charles Wiebel's online source on K-Theory he indicates if you show the kernel of the map $\mathbb{Z}\Pi \to \mathbb{Z}$ is isomorphic to the ideal generated by $\xi -1$ then it is a Milnor square. Why is this true?
Next, in Milnor's proof of Rim's theorem he says "Since $j_{2_*}: K_0\mathbb{Z} \to K_0\mathbb{F}_p$ is clearly an isomorphism it will suffice to verify $j_{1_*} : K_1\mathbb{Z}[\xi] \to K_1\mathbb{F}_p$ is surjective. Why is this enough? I find this rather confusing.
(note: $j_{2_*}$ is the image of $j_2$ under the functor $K_0$ taking rings to abelian groups, $j_{1_*}$ is the image of $j_1$ of $K_1$)