Let $k$ be a field and let $s$ and $t$ be some transcendental indeterminates. This question was originally inspired by an exercise asking me to describe the product scheme of $\text{Spec }k(s)$ with $\text{Spec }k(t)$. This obviously comes down to describing the tensor product $k(s) \otimes_{k} k(t)$. Intuitively I would expect this to be $k(s, t)$, but I am aware that this is not the case. I haven't really come across tensor products of this form before, and am running a blank on familiar identities I could use to describe this.
Is anyone able to give me an idea of what this tensor product looks like, or better yet, a commutative algebra resource that actually covers this kind of thing in detail? It seems like most commutative algebra texts stay away from any transcendental extensions, and and field theory books assume it, so I'm not actually able to find it anywhere.