Purely Inseparable Decompositions

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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.

Dedekind Rings: Factorisation and Ideal Class Group

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

Hensel's Lemma and Field Theory

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

Ramification and Structure of Finite Extensions

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

Finite Degree Extensions

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

Dimension Theory of Rings

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

Ramification Theory: Residual Fields and Discriminant Ideal

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

Galois Theory: Extensions and Residual Fields

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

Algebraic Extensions and Decomposition of Fok [x]

Covers homomorphisms, algebraic extensions, cutting, splitting, and separable elements in Fok [x].

Algebraic Closure of Qp

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