Lectures related to Field extension

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

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

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

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

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

Covers the approximation of functions in Sobolev spaces using smooth functions.

Covers the extension of fields, defining the norm of an element from one field to another, and introducing the p-adic valuation.

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

Covers matrix calculations, basis change, field extensions, complex numbers modulus, and polar decomposition.

Explores the definition of Q_p and the completion of Q,1(p) to form Qp.

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

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

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

Covers the Galois correspondence, relating subgroups to intermediate fields.

Covers magnetic fields, Ampère's law, and magnetic dipoles with examples and illustrations.

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Explores homomorphisms in topology and delves into Galois theory.

Covers stochastic integrals, emphasizing isometry and continuity properties in martingales and different spaces.