Proposition

Prove the Geometric Extension Theorem using the Extension Theorem and Lemma 1. (Ideals, Varieties, and Algorithms)

Solution

First, we will show that $\pi_1(V) \cup (V(c_1, \cdots, c_s) \cap V(I_1)) \subset V(I_1)$.

• $\pi_1(V) \subset V(I_1)$.
  • Lemma 1 on P.129 (Ideals, Varieties, and Algorithms)
• $(V(c_1, \cdots, c_s) \cap V(I_1)) \subset V(I_1)$.
  • This is trivial.

Next, we will show that $V(I_1) \subset \pi_1(V) \cup (V(c_1, \cdots, c_s) \cap V(I_1))$. Let $(a_2, \cdots, a_n) \in V(I_1) \setminus \pi_1(V)$. If no such element exists, we are done. Since $(a_2, \cdots, a_n) \in V(I_1)$, $(a_2, \cdots, a_n)$ is a partial solution. Since $(a_2, \cdots, a_n) \notin \pi_1(V)$, $\forall a_1 \in k, (a_1, \cdots, a_n) \notin V$. By taking the contrapositive of the Extension Theorem, $(a_2, \cdots, a_n) \in V(c_1, \cdots, c_n)$.