When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the circle that induces the identity on the circle. From this lemma, you easily deduce the Brouwer fixed point Theorem for the circle.
This was (for me) one of this "WOOOOW" moments where you realize that abstract constructions and some seemingly innocuous functorial lemmas may yield striking results (especially as I knew a quite long and complicated proof of Brouwer Theorem in dimension 2 before taking this topology class).
I was wondering if there exist (+ reference if they do) similarly "cute" applications of the construction of the étale fundamental group in Algebraic Geometry. Of course "cute" is not well-defined and may vary for each one of us, but existence of fixed points for the Frobenius morphism would I find especially cute. Any other relatively elementary result related to algebraic geometry over fields of positive characteristic will be appreciated!
Edit : I am obviously curious of any application of the étale fundamental group endowed with the aforementioned "WOOOW feeling". However, I'd be really interested in examples I could explain to smart grad students who are taking a first (but relatively advanced) course in Algebraic Geometry.
Best Answer
Using the étale fundamental group one can construct an injective group homomorphism
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$
which is canonical in the sense that there are no choices involved in its construction (once the algebraic closure is fixed), $\operatorname{Out}$ refers to the outer automorphism group and $\widehat{F_2}$ to the profinite completion of the free group on two letters.
As for your example of a "WOOOOW" moment, this statement no longer contains the étale fundamental group in its statement, even though it's vital for the construction.
The fact that the absolute Galois group of the rationals is canonically a subgroup of the outer automorphisms of a (profinite-) free group is completely non-obvious. (Try to prove it from scratch...)
One can try to determine the image of this map. This leads to the profinite Grothendieck-Teichmueller group, which sometimes is conjectured to be isomorphic to $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
The étale fundamental group enters because one considers étale coverings of the projective line minus three points (i.e. the affine line minus two points). Over an algebraically closed field of characteristic zero, this has the étale fundamental group $\widehat{F_2}$ (think of the loops around two of the removed points as its generators). But now study the projective line minus three points over the rationals instead of their algebraic closure, this makes the Galois group of the rationals enter.
The projective line minus three points enters because by a construction due to Belyi, every algebraic curve which can be defined over a finite extension of the rationals can be realized as an étale covering of the projective line minus three points (this is an if and only if). The idea is to modify a map to the projective line so that its ramification gets concentrated in only three points.
You will find the rest of the story for example in surveys by Leila Schneps on the Grothendieck-Teichmueller theory.