Absolute Galois groups viewed from small quotients and the Bloch--Kato conjecture

The second author fondly remembers how a number of years ago Paulo Ribenboim helped him to escape to the West and immediately upon his arrival welcomed him with beautiful lectures on the Galois group of the Pythagorean closure of Q. Ribenboim’s lectures, writings, and research have influenced us strongly, and in particular this paper reflects his influence on the choice of topics and our way of thinking about them. The paper is a selective survey of results on small quotients of absolute Galois groups and their relations with the Bloch–Kato conjecture. It is by no means a comprehensive historical survey. Instead, it focuses only on some selective topics from the work of the authors and their collaborators.