Exercise 1, § 10

Let $\mathbf{K}$ be a field and P the prime subfield of $\mathbf{K}$; the absolute dimension of K is the number dim.al, K if P is of characteristic $p > 0$ and the number dim.al_p K + 1 if P is of characteristic 0. Let $v$ be a valuation on $\mathbf{K}$, hits height, $r$ its rational rank and $k$ its residue field.

(a) Suppose that the absolute dimension $n$ of $K$ is finite. Then, if $s$ is the absolute dimension $k, r + s \leq n$.

(b) Suppose further that $K$ is a finitely generated extension of $P$. Then, if $r + s = n, k$ is a finitely generated extension of its prime field and the order group of $v$ is isomorphic to $\mathbf{Z}^r$; if $h + s = n, k$ is a finitely generated extension of its prime field and the order group of $v$ is isomorphic to $\mathbf{Z}^r$ ordered lexicographically; finally, if $s = n - 1, v$ is a discrete valuation and $k$ is a finitely generated extension of its prime field.