Hilbert's weak Nullstellensatz
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Revision 336
7/8/2026, 8:47:10 AM · Ancient Tree
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If $k$ is an [[Algebraically closed field|algebraically closed field]], then every [[Maximal ideal|maximal ideal]] of the [[Polynomial ring|ring of polynomials]] $k[x_{1},\ldots,x_{n}]$ is an [[Ideal generated by polynomials|ideal generated by polynomials]], of the form $(x_{1}-a_{1},\ldots,x_{n}-a_{n})$ for some point $(a_{1},\ldots,a_{n})$ in the [[Affine space over a field|affine space]] $\mathbb{A}^{n}_{k}$.Revision 335
7/8/2026, 8:44:26 AM · Ancient Tree
Concept edited
Compare with revision 334No text changes
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If $k$ is an [[Algebraically closed field|algebraically closed field]], then every [[Maximal ideal|maximal ideal]] of the [[Polynomial ring|ring of polynomials]] $k[x_{1},\ldots,x_{n}]$ is an [[Ideal generated by polynomials|ideal generated by polynomials]], of the form $(x_{1}-a_{1},\ldots,x_{n}-a_{n})$ for some point $(a_{1},\ldots,a_{n})$ in the [[Affine space over a field|affine space]] $\mathbb{A}^{n}_{k}$.Revision 334
7/8/2026, 8:43:53 AM · Ancient Tree
Concept edited
Compare with revision 3322 changed lines
1
If $k$ is an [[Algebraically closed field|algebraically closed field]], then every [[maximal ideal|maximal ideal]] of the [[Polynomial ring|ring of polynomials]] $k[x_{1},\ldots,x_{n}]$ is an [[Ideal generated by polynomials|ideal generated by polynomials]], of the form $(x_{1}-a_{1},\ldots,x_{n}-a_{n})$ for some point $(a_{1},\ldots,a_{n})$ in the [[Affine space over a field|affine space]] $\mathbb{A}^{n}_{k}$.1
If $k$ is an [[Algebraically closed field|algebraically closed field]], then every [[Maximal ideal|maximal ideal]] of the [[Polynomial ring|ring of polynomials]] $k[x_{1},\ldots,x_{n}]$ is an [[Ideal generated by polynomials|ideal generated by polynomials]], of the form $(x_{1}-a_{1},\ldots,x_{n}-a_{n})$ for some point $(a_{1},\ldots,a_{n})$ in the [[Affine space over a field|affine space]] $\mathbb{A}^{n}_{k}$.Revision 332
7/8/2026, 8:42:53 AM · Ancient Tree
Concept created
If $k$ is an [[Algebraically closed field|algebraically closed field]], then every [[maximal ideal|maximal ideal]] of the [[Polynomial ring|ring of polynomials]] $k[x_{1},\ldots,x_{n}]$ is an [[Ideal generated by polynomials|ideal generated by polynomials]], of the form $(x_{1}-a_{1},\ldots,x_{n}-a_{n})$ for some point $(a_{1},\ldots,a_{n})$ in the [[Affine space over a field|affine space]] $\mathbb{A}^{n}_{k}$.