Whenever I read anything about valuations or things related to them (such as local fields) extensions always occupy a prominent position and a huge amount of effort is expended to derive results about such extensions. I have a little understanding of valuations and I think I have some understanding about how they can be helpful, but I don't understand why so soon after valuations are introduced everything is ignored except for extensions. 



Why are extensions so heavily emphasized in valuation theory?
I don't find your claim that "so soon after valuations are introduced everything is ignored except for extensions" to be accurate. See for instance my notes on valuation theory (as part of a second graduate course on number theory) [here](http://alpha.math.uga.edu/%7Epete/MATH8410.html). Extensions of valuations play a prominent role there, but it is not as if they are the only things discussed. (I think by the way that the same is more or less true for every text on valuation theory I've seen. I picked my notes because I know what's in them and it's easy for me to point to them, not because they are especially exemplary in any way.)


But if I may adjust your question to "Why is it so important to study extensions of valuations?" I can try to give an answer. I think it is helpful to think of things in terms of number theory. (Well, I would, wouldn't I?) If $L/K$ is a
(finite!) extension of number fields, then one of the fundamental problems in algebraic number theory is to understand how the prime ideals in the ring of integers $\mathbb{Z}\_K$ of $K$ decompose when pushed forward to ideals in the ring $\mathbb{Z}\_L$. This theme is relentlessly pursued: Hilbert's ramification theory, Cebotarev density, class field theory...Let's accept that this is really important!


The question of extension of valuations is a precise analogue and generalization of this question. Namely, if $R$ is a Dedekind domain with fraction field $K$, $L$ is a finite field extension of $K$ and
$S$ is the integral closure of $R$ in $L$ (again a Dedekind domain, by the Krull-Akizuki Theorem), then to each nonzero prime ideal $\mathfrak{p}$ of $R$ we can attach the $\mathfrak{p}$-adic valuation $v\_{\mathfrak{p}}$ which assigns to $x \in K^{\times}$ the exponent to which $\mathfrak{p}$ appears in the
factorization of the fractional ideal $xR$ into primes. Then there is a bijective correspondence between the valuations on $L$ extending $v\_{\mathfrak{p}}$ and the primes of $S$ lying over $\mathfrak{p}$. This take on the classical number theory is useful even in studying the classical case: for instance, an especially clean version of "Kummer's criterion" for how a prime factors is to look at the direct factor decomposition of the algebra $L \otimes\_K K\_{\mathfrak{p}}$. In other words, the valuation-theoretic perspective (and the business with completions, which is certainly another really important part of valuation theory!) allows us to get the splitting information we want not from the *ring* $S$ but from the *field* $L$. This is nice because the classical Kummer criterion has problems with a finite number of primes, owing to the fact that $S$ need not be monogenic as an $R$-module. There are no exceptions to the valuation-theoretic analogue.


And from here one gets interested in the problem of extension of valuations for its own sake. One could equally well give an answer in terms of algebraic geometry: if $K$ is a $k$-algebra, then the set of valuations on $K$ which are trivial on $k$ is the **Zariski Riemann space** of $K$. When $K/k$ is finitely generated, this space has a lot of information about the various models of $K$, i.e., $k$-varieties with function field $K$. When $K$ has transcendence degree $1$ the valuations are precisely the closed points on the nonsingular model. In higher dimensions things are more complicated but also more interesting. So then if I have a finite field extension $L/K$, then looking at the extension problem is asking for information about a corresponding dominant rational map on algebraic varieties.


It is certainly useful to know that any non-Archimedean valuation can be extended to any field extension. For instance, Paul Monsky famously and brilliantly proved that it is not possible to dissect a square into an odd number of triangles all of the same area: see [here](http://alpha.math.uga.edu/%7Epete/Monsky.pdf). The "technology" of his argument is the fact that the $2$-adic valuation on $\mathbb{Q}$ extends to a valuation on $\mathbb{R}$. To the best of my knowledge, no one has ever given a proof of Monsky's theorem that does not use some form of this fact on extensions of valuations!


