Algebraic subsets can be decomposed

Last updated January 27, 2022

Theorem. If $V\subseteq\mathbb{A}^n$ is an algebraic subset, then $V$ can be reduced into finitely many irreducible components: $$ V=V_1\cup \cdots\cup V_n. $$ Such a decomposition is unique up to permutation if $V_i\not\subset V_j$ for any $i\neq j$.