TITLE: What makes a theorem "fundamental"?
QUESTION [111 upvotes]: I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems).
Wikipedia claims:

The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.

So I am left wondering: what is the main criterion for a theorem to be considered fundamental?
Why, for instance, is there no "Fundamental Theorem of Complex Analysis" (say, de Moivre's formula, for example), or "Fundamental Theorem of Euclidean Geometry" (e.g. Pythagoras' theorem)?
Does anyone have a source of a more-rigourous definition of "fundamental", in this context?

REPLY [15 votes]: Just a little addendum after @Adam Hughes' nice answer.
Let $(\Omega, \mathcal F, P)$ be a probability space. The information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$) is given by $I(A)=f(P(A))$, denoting by $f$ a real function on $[0,1]$.
Then one has the

Fundamental theorem of information (FTI)
The Shannon entropy $f(x):=x\ln \frac{1}{x}+(1-x)\ln\frac{1}{1-x}$
is the unique continuous solution of the fundamental equation of information (FEI)
$$f(x)+(1-x)f\left(\frac{y}{1-x}\right)=f(y)+(1-y)f\left(\frac{x}{1-y}\right) ~~(*)$$
on  $D:=\{(x,y)~|~ x\in [0,1), y\in[0,1), x+y\leq 1\}$.

The Shannon entropy plays a fundamental role in information theory, information geometry (it even generates Kullback Leibler divergences), and in many applications. There exists even an analogue of the FEI in number theory; for the whole exposition I refer to
Kontsevich, The $1\frac{1}{2}$-logarithm. Composition Math. 130 (2002).
Apart from applications et similia, I believe that it is the formulation of the FTI via functional equations that makes it interesting: after all, many important functions in mathematics are defined as solutions of functional equations (in primis the exponential function): the Shannon entropy becomes one of them.