UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 12

Problem:

Solution:

Part (a) is simple.

A point is in the variety if it is a root.

For part (b).

$g \in I(V(f))  \Leftrightarrow g(a_i) = 0 \text{ }, \forall i \Leftrightarrow g = f_{\text{red}}h \Leftrightarrow g \in \langle f_{\text{red}} \rangle$