Serre's result GAGA shows that when doing complex projective algebraic geometry, it doesn't matter whether one is working algebraically or analytically. Among the consequences of this result is the fact that every finite group is a Galois group over the field C(x), and in fact that the absolute Galois group of C(x) is free. These ideas lead to stronger and more general results in Galois theory, if one extends the notion of "analysis" to include analytic constructions over fields like the p-adics or Laurent series fields.