Dummit and Foote Page :$544$ Proposition $30$

Question

I have a little bit confusion in Dummit and Foote Book on the Page No $:544$

Proposition $30$: For any field $F$ there exist an algebraically closed field $K$ containing $F$

Proof :

My confusion : It is given that $$ g_(x_1,x_2,..,x_n,x_{n+1}....,x_m)f_1(x_1) +.......+g_n(x_1,x_2,..,x_n,x_{n+1}...,x_m)f_n(x_n)=1$$

Now if $x_{n+1}=....=x_m=0$ then $$ g_(x_1,x_2,..,x_n,0....,0)f_1(a_1) +.......+g_n(x_1,x_2,..,0,0...,0)f_n(a_n)=0$$

Here im not getting that why $ g_n(x_1,x_2,..,x_n,0....,0)=0?$

Answer 1

The proof is not claiming that $g_i(x_1,\ldots,x_n,0,\ldots,0)=0$.

Its point is that $f_i(x_i)$ is $0$, because the $x_i=\alpha_i$s were chosen to make this true.

So no matter what the value of $g_i(\cdots)$ is, it will be killed when you multiply it by $f_i(x_i)$.