The book is an undergraduate textbook on abstract algebra, beginning with the theories of rings and groups. As this is the first really abstract material students meet, the pace here is gentle, and the basic concepts of subring, homomorphism, ideal, etc are developed in detail. Later, as students gain confidence with abstractions, they are led to further developments in group and ring theory (simple groups and extensions, Noetherian rings, an outline of universal algebra, lattices and categories) and to applications such as Galois theory and coding theory. There is also a chapter outlining the construction of the number systems from scratch and proving in three different ways that transcendental numbers exist.