Difficulty in understanding the following passage from the book, Inverse Galois Theory by G. Malle I am reading Inverse Galois Theory by Malle and Matzat on my own. 
While reading, I came across the following. I have studied Commutative Algebra and Galois Theory, but I have never seen this terminology. Can you kindly explain what he means? 
Precisely speaking, what does he mean by "prime divisors of the function field", "valuation ideals", "ramified only at prime divisors?"
Thanks in advance.

 A: One of the main points in algebraic geometry is a type of "duality" between spaces and rings of functions. Under this duality, every space locally has a ring of functions (eg the ring of functions of $\text{Spec }A$ is just $A$), and a point of $\text{Spec }A$ corresponds to a prime ideal of $A$.
If $A$ is an integral domain, then let $K$ be its fraction field. E.g., $A = k[x]$, and $K = k(x)$. Thus, $K$ is the (residue field of the) 'generic point' of $A$, and while in a sense $K$ doesn't tell you what $A$ is - for example, $K$ is also the fraction field of $k[x,x^{-1}]$, or even $k[x^{-1}]$. However, in a sense this is because $\text{Spec }A$ isn't the whole picture. Indeed, from $K$, one can recover $\mathbb{P}^1_k$, inside which $\text{Spec }A,\text{Spec }k[x,x^{-1}],\text{Spec }k[x^{-1}]$ all sit.
How do we do this? Well this is explained in I.6 of Hartshorne's Algebraic Geometry. Essentially the procedure is this: Given an algebraically closed field $k$, and a finitely generated extension $K/k$ of transcendence degree 1, let $C_K$ be the set of all discrete valuation subrings of $K$. This set $C_K$ is called an "abstract nonsingular curve", and one can put a scheme structure on $C_K$ to realize it as a proper algebraic curve whose function field is $K$.
Attached to each discrete valuation subring $O\in C_K$ is a corresponding valuation on $v_O : K\rightarrow\mathbb{Z}\cup\{\infty\}$, which can be made into an absolute value $|x|_O := 2^{-v_D(x)}$ (or using any real number $>1$ in place of 2). For such a field $K$, the absolute values $\{|\cdot|_O : O\in C_K\}$ are a complete set of pairwise inequivalent absolute values on $K$, where two absolute values are equivalent if and only if they define the same metric topology on $K$.
Anyway, at this point you might have an idea of what he means.
His $\mathbb{P}(K/\mathbb{C})$ is just the "abstract nonsingular curve" $C_K$ - note that to specify a discrete valuation subring of $K$ is equivalent to specifying its maximal ideal as a subset of $K$. I would guess that a prime divisor of a function field $K$ is just a prime divisor on the corresponding nonsingular curve $C_K$, ie, a closed point of $C_K$, or equivalently a discrete valuation subring of $K$. The valuation ideals of $K$ should just be the maximal ideals of the discrete valuation subrings.
Also, I should say that there is an equivalence of categories between the category of nonsingular curves over $k$ and finite morphisms (over $k$), and finitely generated transcendence deg 1 extensions $K$ of $k$, and inclusions of fields inducing the identity on $k$. The functor goes from the first to the second by taking "function fields". 
At this point, you can think of ramification along prime divisors in terms of ramification of curves along (closed) points. 
If you've seen some number theory, then ramification can also be understood in terms of fields. For example, given a map of curves $f : X\rightarrow Y$ ramified above a point $y\in Y$, let $x\in f^{-1}(y)$, and let $K_X,K_Y$ be the function fields of $X,Y$, then $x,y$ correspond to discrete valuation subrings $O_x,O_y$ of $K_X,K_Y$, defining absolute values $|\cdot|_x,|\cdot|_y$ on  $K_X,K_Y$. Let $\hat{O}_x,\hat{O}_y$ denote the completion of $O_x,O_y$ w.r.t. their maximal ideals, and let $K_x := \text{Frac }\hat{O}_x$ and $K_y := \text{Frac }\hat{O}_y$. Then, $K_x,K_y$ contain $K_X,K_Y$, and can be identified with the completion of $K_X,K_Y$ as metric spaces w.r.t. the absolute values $|\cdot|_x,|\cdot|_y$ (the resulting completions naturally inherits the structure of a field). $K_x,K_y$ are called local fields at $x$ or $y$. The fact that $f(x) = y$ implies that we have a local ring homomorphism $O_y\rightarrow O_x$, whence a map $\hat{O}_y\rightarrow\hat{O}_x$, whence a field extension $K_x/K_y$. In this case, we can scale the absolute value $|\cdot|_x$ on $K_x$ so that it becomes an extension of $|\cdot|_y$, and we can say that the extension is unramified (ie, at $x$, or above $y$) if any of the following equivalent conditions hold:


*

*The maximal ideal $m_y$ of $O_y$ generates the maximal ideal of $O_x$.

*The absolute value $|\cdot|_x$ has the same image as its restriction to $K_y$.

