This is actually quite a famous problem, found here or here on SE.
Let $A$ and $B$ be finitely generated $k$-algebras over a field $k$. Show that $$\dim(A \otimes_k B) = \dim(A) + \dim(B).$$
Here, $\dim$ denotes the Krull dimension.
I actually almost have a complete solution, but fail to show that $k[X_1, \dots, X_n, Y_1, \dots, Y_m] \to A \otimes_k B$ is an injection where we already know that $k[X_1, \dots, X_n] \hookrightarrow A$ and $k[Y_1, \dots, Y_m] \hookrightarrow B$ are finite ring embeddings by Noether normalisation.
What I tried is the following: We know that $k[X_1, \dots, X_n]$ is finite and hence flat as a $k$-module. Thus we can tensor the injection $k[Y_1, \dots, Y_m] \hookrightarrow B$ with $k[X_1, \dots, X_n]$ to obtain the injection $$k[X_1, \dots, X_n, Y_1, \dots, Y_m] \cong k[X_1, \dots, X_n] \otimes_k k[Y_1, \dots, Y_m] \hookrightarrow k[X_1, \dots, X_n] \otimes_k B.$$ But how do I show that $$k[X_1, \dots, X_n] \otimes_k B \hookrightarrow A \otimes_k B$$ is an injection? As far as I know, $B$ is not necessarily flat, so we cannot do the same trick as above.