Lectures related to Algebras and Field Extensions

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Explores homomorphisms in topology and delves into Galois theory.

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

Covers the construction and universal properties of pushouts in group theory.

Explores subspaces in linear algebra and transformations, including kernels and images of linear transformations.

Covers the Galois correspondence, relating subgroups to intermediate fields.

Explores Cayley tables, group operations, homomorphisms, Lie groups, and differentiable groups.

Covers the fundamentals of polynomials, endomorphisms, division, roots, matrices, and algebraic homomorphisms.

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Explores Schur's lemma and its applications in representations of an associative algebra over an algebraically closed field.

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

Covers the structure of finite dimensional algebras and the characterization of semisimple algebras.

Covers tensor algebras, symmetric and exterior algebras, and tensor products of modules.

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.

Covers the density theorem in representation theory, focusing on endomorphisms of simple finite-dimensional modules.

Covers simple modules, endomorphisms, and Schur's lemma in module theory.

Covers the properties of finite dimensional algebras and non-semisimple representations.

Explores congruence relations in rings, principal ideals, ring homomorphisms, and the characteristic of rings.