Vanishing Ideal of Zero Locus of Ideal is Radical

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Theorem

Let $k$ be an algebraically closed field.

Let $n \ge 0$ be a natural number.

Let $k \sqbrk {X_1, \ldots, X_n}$ be the polynomial ring in $n$ variables over $k$.

Let $\mathfrak a \subseteq k \sqbrk {X_1, \ldots, X_n}$ be an ideal.

Then:

$\map I {\map V {\mathfrak a} } = \map \Rad {\mathfrak a}$

where:

$\map V \cdot$ denotes the zero locus

$\map I \cdot$ denotes the vanishing ideal

$\map \Rad \cdot$ denotes the radical

Proof

This is exactly Hilbert's Nullstellensatz.

$\blacksquare$