You know basic concepts from differential geometry: manifolds, tangent spaces, dimension, diffeomorphisms etc. .
Algebraic geometry is the theory that uses geometric insight to discover new results in number theory. We introduce the prime spectrum and etale extensions, which are ringtheoretic counterparts of coverings of manifolds. We introduce the etale fundamental group and use it to give an analogy between prime numbers and knots, reciprocity results and link behavior.
Course notes will be provided.