Knots and Primes

The development of homology theory in topology influenced algebraic number theory. Nakayama and Tate established the theory of Galois cohomology and applied it to give a new proof of class field theory. Motivated by Weil’s conjectures, stating that there is a deep connection between arithmetic properties of an algebraic variety over a finite field and topological properties of the corresponding complex manifold, Grothendieck elaborated the theory of étale topology and introduced the étale fundamental group and étale cohomology for schemes, which have similar properties to the topological fundamental group and singular cohomology, respectively.