# Weak Nullstellensatz

## Theorem

Let $k$ be an algebraically closed field.

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

Let $k \left[{x_1,\ldots, x_n}\right]$ be the polynomial ring in $n$ variables over $k$.

Let $I \subseteq k \left[{x_1,\ldots, x_n}\right]$ be an ideal.

The following are equivalent:

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