The aim of the course is to give an elementary introduction to Algebraic Number Theory. The course will focus on the study of rings of algebraic integers (the analogue, inside number fields, of the ring of integers Z seen as subfield of Q): within this rings one loses the unique factorization property for elements, but this property is recovered for ideals (for instance, we will understand the origin of the name "ideal"). From an historic perspective, on the one hand the study of rings of algebraic integers led to the development of ring theory, and on the other hand it has been motivated by the "correction" of what we can call the "best wrong proof" of Fermat's Last Theorem!
Indeed, at the end of the course we will see Kummer's proof of FLT for regular primes.