I am trying to do exercise 3.1.5 in Liu's book which asks the following.
Let $X,Y$ be algebraic varieties over a field $k$. Show that $\dim(X\times_k Y)=\dim X\, + \dim Y$.
The hint is to use proposition $2.19$ which says that the dimension of an integral algebraic variety is its transcendence degree. While I thought this would be straightforward, once I do the calculation I end up trying to compute the dimension of $\text{Spec}(\mathcal{O}_X(X)\otimes_k \mathcal{O}_Y(Y))$ and I'm not sure how to apply the the theorem on the transcendence degree. My intuition is that I need to use Noether normalization, but I don't see how this gives me what I want. Could someone help me clear this up? Also, can I actually use the theorem since it seems to require integrality (which I thought was baked into the definition of algebraic variety, but maybe not). I saw there were other similar questions but none were quite like this one. If I got it wrong please don't hesitate to redirect me to the right question.