Suppose $K=\mathbb{Q}(X\_1,\dots,X\_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be discrete? 


If not, is there a theorem that shows that such a rank $1$ valuation must be discrete?



Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals
Example: $val(X\_1^{e\_1}X\_2^{e\_2})=e\_1 + e\_2 \sqrt{2}$.


