In Bjorn Poonen's book Rational Points on Varieties he says that Fermat Descent is an example of cohomology. There is also a book by Soulé. Even Wikipedia mentions this with no further explanation. [1]
Many techniques have been used to decide whether a variety over a number field has a rational point. Some generalize Fermat’s method of infinite descent, some use quadratic reciprocity, and others appear at first sight to be ad hoc. But over the past few decades, it was discovered that nearly all of these techniques could be understood as applications of just two cohomological obstructions, the étale-Brauer obstruction and the descent obstruction. Moreover, while this book was being written, it was proved that the étale-Brauer obstruction v and the descent obstruction are equivalent! The topics in this book build up to a full explanation of this “grand unified theory” of obstructions
There are many explanations of Fermat Descent both on MathOverflow and the junior site Math.Exchange. I only know two very well:
It seems these irrationality proofs and sum of squares proofs use a cohomology theory at least implicitly. Could someone explain how that is done or explain the type of question that can be answered this way?
Neither Poonen nor Soulé really explain how these elementary problems get encoded into the advanced language. What are the cohomology groups being used here? How do they solve the problem?
