“Nullstellensatz”的概念、定义、翻译、参考文献-科学参考

Nullstellensatz

Mathematics

The following important result in commutative algebra due to Hilbert. Let F be an algebraically closed field (such as ℂ) and n be a positive integer. For an ideal J in the polynomial ring F[x₁,…,x_n] and for a subset S of Fⁿ we may define

$V (J) = {x \in F^{n}; | f (x) = 0 for all f \in J}; I (S) = {f \in F [x_{1}, \dots, x_{n}]; | f (x) = 0 for all x \in S} .$

Then $I (V (J)) = \sqrt{J}$ , the radical of J. By comparison $V (I (S)) = \bar{S}$ , the closure of S in the Zariski topology.

单词	Nullstellensatz
释义	Nullstellensatz Mathematics The following important result in commutative algebra due to Hilbert. Let F be an algebraically closed field (such as ℂ) and n be a positive integer. For an ideal J in the polynomial ring F[x₁,…,x_n] and for a subset S of Fⁿ we may define $V (J) = {x \in F^{n}; \| f (x) = 0 for all f \in J}; I (S) = {f \in F [x_{1}, \dots, x_{n}]; \| f (x) = 0 for all x \in S} .$ Then $I (V (J)) = \sqrt{J}$ , the radical of J. By comparison $V (I (S)) = \bar{S}$ , the closure of S in the Zariski topology.
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