Ramification and Structure of Finite Extensions

Related lectures (34)

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

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

Explores the Localization Theorem in Dedekind rings, isomorphism induced by injection, and ramification in field theory.

Explores ramified extensions and Eisenstein polynomials, showcasing their applications in mathematical contexts.

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

Explores ramification theory, residue fields, Galois extensions, and decomposition groups in algebraic number theory.

Explores the construction and properties of finite fields, including irreducible polynomials and the Chinese Remainder Theorem.

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

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

Covers principal ideals, ring homomorphisms, and more in commutative rings and fields.

Covers the theory and operations related to polynomials, including ideals, minimal polynomials, irreducibility, and factorization.

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

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

Explores ring homomorphisms, bilateral ideals, ring features, and ideal operations.

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