Hensel's Lemma and Field Theory

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Related lectures (34)

Topology: Homomorphisms and Galois Theory

Explores homomorphisms in topology and delves into Galois theory.

Galois Correspondence

Covers the Galois correspondence, relating subgroups to intermediate fields.

Intermediate Coatings: Revisiting Galoisian Correspondence

Revisits the Galoisian correspondence and explores group actions in intermediate coatings.

Elliptic Curve Cryptography: Galois Fields

Explores Galois fields, elliptic curve cryptography, arithmetic operations, group structure, and practical examples in cryptography.

Galois Fields and Elliptic Curves

Introduces Galois fields, elliptic curves, factorization algorithms, and the discrete logarithm problem in cryptography.

Galois Theory of Qp Log in to Mediaspace to watch this video

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

Galois Theory Fundamentals Log in to Mediaspace to watch this video

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.

Algebraic Closure of Qp Log in to Mediaspace to watch this video

Covers the algebraic closure of Qp and the definition of p-adic complex numbers, exploring roots' continuous dependence on coefficients.

Norm Extension in Finite Fields Log in to Mediaspace to watch this video

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

Purely Inseparable Decompositions Log in to Mediaspace to watch this video

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

Galois Theory: The Galois Correspondence Log in to Mediaspace to watch this video

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

Galois Theory: Extensions and Residual Fields Log in to Mediaspace to watch this video

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

Ramification and Structure of Finite Extensions Log in to Mediaspace to watch this video

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

Galois Theory: Solvability and Radical Extensions Log in to Mediaspace to watch this video

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

Finite Degree Extensions Log in to Mediaspace to watch this video

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

Galois Theory: Recap and Transitivity Log in to Mediaspace to watch this video

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

Decomposition & Inertia: Group Actions and Galois Theory Log in to Mediaspace to watch this video

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

Galois Theory: Dedekind Rings Log in to Mediaspace to watch this video

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

Dedekind Rings: Factorisation and Ideal Class Group Log in to Mediaspace to watch this video

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

Dedekind Rings: Theory and Applications Log in to Mediaspace to watch this video

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

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