Lectures related to Galois Theory: Recap and Transitivity

Ramification and Structure of Finite Extensions

Explores ramification and structure of finite extensions of Qp, including unramified extensions and Galois properties.

Topology: Homomorphisms and Galois Theory

Explores homomorphisms in topology and delves into Galois theory.

Explores Galois theory fundamentals, including separable elements, decomposition fields, and Galois groups, emphasizing the importance of finite degree extensions and the structure of Galois extensions.

Finite Degree Extensions

Covers the concept of finite degree extensions in Galois theory, focusing on separable extensions.

Galois Theory: The Galois Correspondence

Explores the Galois correspondence and solvability by radicals in polynomial equations.

Galois Theory: Extensions and Residual Fields

Explores Galois theory, unramified primes, roots of polynomials, and finite residual extensions.

Classification of Extensions

Explores the classification of extensions in group theory, emphasizing split extensions and semi-direct products.

Galois Theory of Qp

Explores the Galois theory of Qp, covering algebraic extensions, inertia groups, and cyclic properties.

Decomposition & Inertia: Group Actions and Galois Theory

Explores decomposition groups, inertia subgroups, Galois theory, unramified primes, and cyclotomic fields in group actions and field extensions.

Galois Theory: Solvability and Radical Extensions

Explores solvability by radicals in Galois theory and the Galois/Abel criterion for solvability.

Purely Inseparable Decompositions

Explores purely inseparable decompositions, Galois property, and algebraic closures.

Hensel's Lemma and Field Theory

Covers the proof of Hensel's Lemma and a review of field theory, including Newton's approximation and p-adic complex numbers.

Galois Correspondence

Covers the Galois correspondence, relating subgroups to intermediate fields.

Dimension Theory of Rings

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Algebraic Extensions and Decomposition of Fok [x]

Covers homomorphisms, algebraic extensions, cutting, splitting, and separable elements in Fok [x].

Norm Extension in Finite Fields

Covers the uniqueness of norm extension in finite fields and the construction of norms on finite extensions of Qp.

Algebras and Field Extensions

Introduces algebras over a field, k-linear endomorphisms, and commutative algebras.

Ramification Theory: Residual Fields and Discriminant Ideal

Explores ramification theory, residual fields, and discriminant ideals in algebraic number theory.

Dedekind Rings: Theory and Applications

Explores Dedekind rings, integral closure, factorization of ideals, and Gauss' Lemma.

Galois Theory: Dedekind Rings

Explores Galois theory with a focus on Dedekind rings and their unique factorization of fractional ideals.