# proof of Hilbert’s Nullstellensatz

Let $K$ be an algebraically closed field, let $n\ge 0$, and let $I$ be an ideal of the polynomial ring^{} $K[{x}_{1},\mathrm{\dots},{x}_{n}]$. Let $f\in K[{x}_{1},\mathrm{\dots},{x}_{n}]$ be
a polynomial^{} with the property that

$$f({a}_{1},\mathrm{\dots},{a}_{n})=0\text{for all}({a}_{1},\mathrm{\dots},{a}_{n})\in V(I).$$ |

Suppose that ${f}^{r}\notin I$ for all $r>0$; in particular, $I$ is strictly smaller than $K[{x}_{1},\mathrm{\dots},{x}_{n}]$ and $f\ne 0$. Consider the ring

$$R=K[{x}_{1},\mathrm{\dots},{x}_{n},1/f]\subset K({x}_{1},\mathrm{\dots},{x}_{n}).$$ |

The $R$-ideal $RI$ is strictly smaller than $R$, since

$$RI=\bigcup _{r=0}^{\mathrm{\infty}}{f}^{-r}I$$ |

does not contain the unit element. Let $y$ be an indeterminate over
$K[{x}_{1},\mathrm{\dots},{x}_{n}]$, and let $J$ be the inverse image of $RI$ under
the homomorphism^{}

$$\varphi :K[{x}_{1},\mathrm{\dots},{x}_{n},y]\to R$$ |

acting as the identity^{} on $K[{x}_{1},\mathrm{\dots},{x}_{n}]$ and sending $y$ to
$1/f$. Then $J$ is strictly smaller than $K[{x}_{1},\mathrm{\dots},{x}_{n},y]$, so
the weak Nullstellensatz gives us an element $({a}_{1},\mathrm{\dots},{a}_{n},b)\in {K}^{n+1}$ such that $g({a}_{1},\mathrm{\dots},{a}_{n},b)=0$ for all $g\in J$. In
particular, we see that $g({a}_{1},\mathrm{\dots},{a}_{n})=0$ for all $g\in I$. Our
assumption^{} on $f$ therefore implies $f({a}_{1},\mathrm{\dots},{a}_{n})=0$. However,
$J$ also contains the element $1-yf$ since $\varphi $ sends this element
to zero. This leads to the following contradiction^{}:

$$0=(1-yf)({a}_{1},\mathrm{\dots},{a}_{n},b)=1-bf({a}_{1},\mathrm{\dots},{a}_{n})=1.$$ |

The assumption that ${f}^{r}\notin I$ for all $r>0$ is therefore false, i.e. there is an $r>0$ with ${f}^{r}\in I$.

Title | proof of Hilbert’s Nullstellensatz |
---|---|

Canonical name | ProofOfHilbertsNullstellensatz |

Date of creation | 2013-03-22 15:27:46 |

Last modified on | 2013-03-22 15:27:46 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 4 |

Author | pbruin (1001) |

Entry type | Proof |

Classification | msc 13A10 |