Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra'' by Mines, Richman, and Ruitenburg

The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become much more precise. We are often surprised to find proofs that are simpler and more elegant than the usual ones. In fact, when one cannot use magic tools as the law of excluded middle, it is necessary to understand what is the true content of a classical proof. Also, usual shortcuts allowed in classical proofs introduce sometimes useless detours. In order to understand clearly a problem, prescience may be a handicap.