Intersection of Zero Loci is Zero Locus

Theorem

Let $k$ be a field.

Let $n \in \N_{>0}$.

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.

Let $\mathbb S \subseteq \powerset A$ be a subset of the power set of $A$.

Then:

$\ds \bigcap _{S \mathop \in \mathbb S} \map V S = \map V {\bigcup \mathbb S}$

where $\map V \cdot$ denotes the zero locus.

Proof

\(\ds x\)

\(\in\)

\(\ds \bigcap_{S \mathop \in \mathbb S} \map V S\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall S \in \mathbb S: \, \)

\(\ds x\)

\(\in\)

\(\ds \map V S\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall S \in \mathbb S: \forall f \in S: \, \)

\(\ds \map f x\)

\(=\)

\(\ds 0\)

Definition of Zero Locus of Set of Polynomials

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall f \in \bigcup \mathbb S: \, \)

\(\ds \map f x\)

\(=\)

\(\ds 0\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds \map V {\bigcup \mathbb S}\)

$\blacksquare$