Lectures in course MATH-317: Algebra V - Galois theory

Integral Domains: Factorisation and Noetherian Rings

Explores factorisation in Principal Ideal Domains and Noetherian rings, emphasizing the integral closure concept and the factorisation of ideals in Dedekind rings.

Galois Theory: The Galois Correspondence

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

Galois Theory: Solvability and Radical Extensions

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

Dedekind Rings: Integral Extensions and Noetherian Rings

Explores Dedekind rings, integral extensions, and noetherian rings in algebraic structures.

Dedekind Rings and Fractional Ideals

Explores Dedekind rings, fractional ideals, integrally closed properties, prime ideal factorization, and the structure of fractional ideals as a commutative group.

Dedekind Rings: Factorisation and Ideal Class Group

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

Decomposition & Inertia: Group Actions and Galois Theory

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

Frobenius Theorems in Number Theory

Explores Frobenius theorems in number theory, ideal class groups, norm properties, and geometry of numbers.

Cyclotomic Extensions: Norms, Ideals, and Primes

Explores cyclotomic extensions, prime numbers, and ideal norms in number theory.

Minkowski's Theorems: Lattices and Volumes

Explores Minkowski's theorems on lattices, volumes, and set comparisons.

Lattice Theory: Minkowski's Theorems

Delves into lattice theory, emphasizing Minkowski's theorems and their implications on lattice structures.

Embeddings of Number Fields

Explores embeddings of number fields, types, signatures, lattices, and determinants.

Number Fields: Embeddings and Ideal Classes

Covers the embeddings of number fields and ideal classes with proofs and examples.

Hermite-Minkowski Theorems: Number Fields and Ideal Classes

Explores Hermite-Minkowski theorems in number fields and ideal classes.

Logarithmic Embedding in Number Fields

Explores the properties and applications of logarithmic embeddings in number fields.

Class Number Formula

Explores the Class Number Formula in number theory, focusing on Lemmas, Proofs, and a significant theorem.

The Class Number Formula: Counting and Lipschitz Principle

Covers the Class Number Formula and a counting problem related to the lattice and Lipschitz principle.

Lipschitz Maps and Compact Domains

Covers Lipschitz maps, compact domains, changing variables, counting problems, and probability of ideals.

Dedekind Function: Analytic Continuation and Euler Product Formula

Covers the Dedekind function, Euler product formula, series convergence, and analytic continuation of logarithmic functions.