I like Courant and Robbins' book: What is Mathematics? It does not have Galois theory but does describe how to use elementary field theory to prove the impossibility of the three classic Greek construction problems. It also has a section on topology and one on the fundamental theorem of arithmetic. There is a clever proof of that theorem without proving first the usual prime divisibility property.

The book Geometry and the Imagination by Hilbert and Cohn - Vossen, also has some nice elementary topology as I recall.

These books were meant to be accessible to the intelligent lay person.