The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and algebraic extensions of these. One might call such fields <b>$0$-dimensional</b>. Then one could say that a field $K$ is <b>$d$-dimensional</b> if it has transcendence degree $d$ over a $0$-dimensional field.

But is there a way to make this less ad hoc? Is there a reason I've never seen any such definition of <b>$0$-dimensional fields</b>? Am I missing something?

To what extent has the classification of abstract fields been considered?