Localization Theorem in Dedekind Rings

This lecture covers the Localization Theorem in Dedekind rings, stating that for a Dedekind ring A and its integral closure B in a finite separable extension of Q, the dimension of B/p.B is equal to the degree of the extension. It also explores the properties of localization at a multiplicative set, the uniqueness of maximal ideals in a PID, and the isomorphism induced by the injection A to Ap. Additionally, it discusses the integral closure of Ap in K, leading to an isomorphism of kp-algebras. The lecture concludes with the concept of ramification in field theory, defining ramified primes in K and unramified extensions in B.