Let $k$ be any field, and take any $0 \not = F \in k [x_1,..., x_n]$. Is it always the case that the Krull dimension of $k[x_1, ..., x_n]/(F)$ is $n-1$?

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If by $k[x_1,..., x_n]$ you denoted a polynomial ring, then this is the case. In fact, $\dim k[x_1,..., x_n]/(F)=\sup\{\dim k[x_1,..., x_n]/P:P\text{ is a minimal prime over }(F)\}$. Since $P$ minimal over $(F)$ its height is one, and then $\dim k[x_1,..., x_n]/P=n-1$.

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