The Ideal of a Set of Points Review

# The Ideal of a Set of Points Review

We will now review some of the recent material regarding the ideal of a set of points.

• On the [[[The Ideal of a Set of Points]] we said that if $K$ is a field and $X \subseteq \mathbb{A}^n(K)$ then the Ideal of $X$ is defined as the ideal:
(1)
\begin{align} \quad I(X) = \{ F \in K[x_1, x_2, ..., x_n] : F(\mathbf{p}) = 0, \: \forall \mathbf{p} \in X \} \end{align}
• That is, the ideal of $X$ is the ideal which contains all polynomials in $K[x_1, x_2, ..., x_n]$ that vanish at every point in $X$. We then verified that $I(X)$ is indeed an ideal in $K[x_1, x_2, ..., x_n]$.
Number Property
1 If $X, Y \subseteq \mathbb{A}^n(K)$ and $X \subseteq Y$ then $I(X) \supseteq I(Y)$.
2 $I(\emptyset) = K[x_1, x_2, ..., x_n]$.
3 If $K$ is an infinite field then $I(\mathbb{A}^n(K)) = (0)$.
• We then looked at some properties which relate $V$ and $I$:
Number Property
4 If $S \subseteq K[x_1, x_2, ..., x_n]$ then $S \subseteq I(V(S))$.
5 If $X \subseteq \mathbb{A}^n(K)$ then $X \subseteq V(I(X))$.
(2)
\begin{align} \quad \mathrm{Rad}(I) = \{ a \in R : a^n \in I, \: \mathrm{for \: some \:} n \in \mathbb{N} \} \end{align}
• We proved that $\mathrm{Rad}(I)$ is indeed an ideal and said that an ideal $I$ is a Radical Ideal if:
(3)
• We then proved a nice result. We said that if $X \subseteq \mathbb{A}^n(K)$ then the ideal of $X$ is a radical ideal, that is: