Ramification Theory: Residual Fields and Discriminant Ideal

This lecture covers the ramification index, residual fields, inertia degree, and the discriminant ideal in the context of algebraic number theory. It explains the concepts of reduced algebras, ramification in field extensions, and the finite set of ramified primes. The instructor discusses the discriminant ideal of a Dedekind ring, separable extensions, and the properties of residue fields. The lecture concludes with the converse hypothesis regarding perfect residue fields and the ramification of primes.