Lectures related to Purely inseparable extension

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

Covers the damped harmonic oscillator in quantum optics, discussing alternative descriptions and extensions like finite temperature.

Dedekind Rings: Factorisation and Ideal Class Group

Explores Dedekind rings, factorisation, ideal class group, heredity, separable extensions, and matrix properties.

Finite Degree Extensions

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

Ramification Theory: Residual Fields and Discriminant Ideal

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

Separable Extensions: Dedekind Rings

Explores separable extensions and Dedekind rings, focusing on coefficients and prime ideals.

Special open sets and epsilonization

Covers the concept of special open sets and epsilonization, discussing unique extensions of derivations and specific value settings.

Extension of Domains

Covers extending domains, leading coefficients, and valid extensions.

Classification of Extensions

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

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Galois Theory: Recap and Transitivity

Covers the recap of Galois theory and emphasizes the transitivity of Galois groups.

Ramification and Structure of Finite Extensions

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

Infinity Category Theory: Extensions and Equivalences

Explores right extensions in infinity category theory and their equivalent definitions.

Damped harmonic oscillator: Introduction and Field Amplitude

Covers the damped harmonic oscillator, introducing the field amplitude and coherent states.

Properties of Finite Dimensional Algebras

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

Differential Equations: Solutions and Extensions

Covers solving differential equations, local and global solutions, and uniqueness.

Galois Theory Fundamentals

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.

Differentiating under the integral sign

Explores differentiating under the integral sign and conditions for differentiation, with examples and extensions to functions on open intervals.

Hilbert Transform: Exercises and Key Lemmas

Covers exercises and key lemmas related to the Hilbert transform.

L-values and Extensions

Explores de Rham theory, L-values, and extensions, including special value formulas and examples related to Hecke characters and modular forms.