Overview of advanced algebra

RAG -- Rigour, Abstraction and Generality is a
crucial aspiration of algebra, but it is also a
major stumbling block. There is not much point
in knowing abstract theory without being able
to use it in calculations. The reductionist
approach in science breaks knowledge up into
elementary steps; but the student who can only
follow an argument line by line cannot hope to
make the creative leap to new horizons.

Algebra is an environment for calculating. We
first have to learn how to use different parts
of algebra as tools to calculate with.
Explicit examples are most relevant for this
purpose, and I will often only dismiss the
ragged connections with a hint or a reference
to the Wikipedia pages.