Es seien m v = { F ∈ K [ V ] ∣ F ( v ) = 0 } {\displaystyle {}{\mathfrak {m}}_{v}={\left\{F\in K[V]\mid F(v)=0\right\}}} und m w = { G ∈ K [ W ] ∣ G ( w ) = 0 } {\displaystyle {}{\mathfrak {m}}_{w}={\left\{G\in K[W]\mid G(w)=0\right\}}} die zu den Vektoren v {\displaystyle {}v} bzw. w {\displaystyle {}w} gehörigen maximalen Ideale. Die Aussage folgt aus
![{\displaystyle {}{\mathfrak {m}}_{v}={\left\{F\in K[V]\mid F(v)=0\right\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f41a8e273b0524eaa5ec5595c54cbc83f3af7207)
![{\displaystyle {}{\mathfrak {m}}_{w}={\left\{G\in K[W]\mid G(w)=0\right\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/462e02dc61cad3d9979b987145fb0e450ad98e95)
![{\displaystyle {}{\begin{aligned}\varphi ^{*}({\mathfrak {m}}_{v})&=\varphi ^{-1}({\mathfrak {m}}_{v})\\&={\left\{G\in K[W]\mid \varphi (G)\in {\mathfrak {m}}_{v}\right\}}\\&={\left\{G\in K[W]\mid \varphi (G)(v)=0\right\}}\\&={\left\{G\in K[W]\mid {\left(G\circ \psi \right)}(v)=0\right\}}\\&={\left\{G\in K[W]\mid G(\psi (v))=0\right\}}\\&={\mathfrak {m}}_{\psi (v)}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a181e1307761baa78d63a4f5b28a2de6cf8b8cea)