# A question about the uniqueness in Hilbert's Nullstellensatz

Suppose $$k$$ is an algebraically closed field. One of the formulations of Hilbert's weak Nullstellensatz is that each maximal ideal in $$k[x_1, \dots , x_n]$$ has the form $$\langle x_1 -a_1 , \dots , x_n - a_n\rangle$$, where $$a = (a_1 , \dots , a_n)$$ is a point in affine space $$k^n$$. Is the point $$a$$ unique?

Yes, assume $$b = (b_1,\dots,b_n)$$ is another point like $$a$$ then: $$\langle x_1-a_1,\dots,x_n-a_n\rangle=\langle x_1-b_1,\dots,x_n-b_n\rangle .$$ So $$x_1 -b_1 = \sum f_i (x_i-a_i) \ \ , \ \exists f_i \in k[x] .$$ Can you explain why this is not possible if $$b_1 \not= a_1$$?
• The $f_i \in k[x]$ or $f_i \in k[x_1, \cdots x_n]$ – Werner Germán Busch May 5 at 2:18