Weak Nullstellensatz

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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: $V(I) = \varnothing$.


Proof