Let $K$ be a field, we say that a function $\nu:K\to \mathbb{Z}_{\geq 0} \cup \lbrace \infty \rbrace$ is a valuation above $K$ if then followings hold for each choose of $x,y \in K$:
1 $\nu(x) = \infty \Leftrightarrow x = 0$;
2 $\nu(xy) = \nu(x) + \nu(y)$;
3 $\nu(x+y) \geq \min\lbrace\nu(x),\nu(y)\rbrace$.
My question is: if we have a field $K$ that cannot be written like the field of fraction of an UFD (I think that $\mathbb{Q}(\sqrt{-5})$ is a field with this property), what can we say about the valuations. I mean
1 Does it always exists a non trivial valuation above $K$?
2 If it exists, what can we say about that, can we describe how the valuations behave? (for further details of "what can we say about that" see below. I consider the Ostrowski theorem sufficiently well describing the how the valuation is "done")
I have an answer in the case in with $K$ IS the field of fraction of an UFD, below I explain the solution in this case.
An observation. If $\alpha$ is a complex algebraic number above $\mathbb{Q}$ (i.e. it exists a polynomial with rational coefficients among which roots there is $\alpha$), then the field $\mathbb{Q}(\alpha)$ coincides with the fields of fractions of the domain $\mathbb{Z}(\alpha)$.
We can now analyse some examples. The $p$-adic valuations are valuations (in the sense of the above definition) above $\mathbb{Q}$.
We can define in an analogous way of the $p$-adic valuations above $\mathbb{Q}$ a family of valuations above a field that is a field of fraction of an unique factorisation domain. Let's see how. Let $A$ be an UFD and $K$ be its field of fractions.
Let $\mathcal{P}$ be the set of prime elements of $A$. Every nonzero element $x$ of $A$ can be written (identifying the invertibles as one elements) in an unique way like
$x = u\prod_{\pi \in \mathcal{P}}\pi^{\gamma_{\pi}}$,
where $u$ is an invertible element of $A$ and the numbers $\gamma_{\pi}$ are non negative integers. Let chose a prime element $\pi$ of $A$. We define the function $\nu_{\pi}:A$, as
$\nu_{\pi}(x) = \gamma_{\pi}$.
We can now extend the domain of the function $\nu_{\pi}$ from $A$ to $K$ defining for each element $a/b \in K$ the value
$\nu_{\pi}(a/b) = \nu_{\pi}(a) - \nu_{\pi}(b)$.
It can be easily be proven that $\nu_{\pi}$ is a valuation.
There is a theorem from Ostrowski that states that every non trivial valuations above $\mathbb{Q}$ must be a $p$-adic valuation for a prime $p \in \mathbb{Z}$ or a $p$-adic valuation multiplied by a rational constant.
I am pretty sure that it can be generalized to a field of fraction of an UFD without problems. So, if the generalization is true, we have a characterization of all valuations above the field of fractions of an UFD.
Origin of the question: I am reading an article form Roberts and Vivaldi (http://www.maths.qmul.ac.uk/~fv/research/Height.pdf) and in the proof of the Theorem 1 at pag. 6 they consider a valuation above a quadratic extension of the completion of the rationals with respect to a $p$-adic norm ($p \in \mathbb{Z}$). I want to know how their valuation is done (and to prove that it exists!). Their situation is different from mine because they have got a valuation above the extension of a completion of $\mathbb{Q}$, while I am working directly with an extension of $\mathbb{Q}$. I am doing this way because I do not know too much about the completion of $\mathbb{Q}$ with respect to some $p$-adic norm beyond the definition and some basic facts. So if you are not able to find a solution to my problem, for me it would the same if you suppose that the algebraic degree of $\alpha$ above $\mathbb{Z}$ is two or you are able to explain why THEIR valuation exists and how it is done.