Is that the breakdown? Euclid's Algorithm, Pythagoras' Theorem, Fermat's Little Theorem are all elementary, but the example upthread of a "perfectoid field" which is a "complete topological field K whose topology is induced by a nondiscrete valuation of rank 1..." is much more abstruse.

There's no system. Abelian groups are a concept that is simple, intuitive, and easy to describe concisely.