Weak Nullstellensatz

Theorem

Let $K$ be an algebraically closed field.

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

Let $K \sqbrk {x_1, \ldots, x_n}$ be the polynomial ring in $n$ variables over $k$.

Let $I \subseteq K \sqbrk {x_1,\ldots, x_n}$ be an ideal.

The following statements are equivalent:

1. $I$ is the unit ideal: $I = (1)$.
2. Its zero-locus is empty set: $\map V I = \O$.

Proof

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