Lectures related to Polynomials on a Field: Properties and Applications

Chinese Remainder Theorem: Rings and Fields

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

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

Polynomials, Division, and Ideals

Explores polynomials, their operations, and the concept of ideals in polynomial rings.

Irreducible Factors and Noetherian Rings

Discusses irreducible factors in rings and the properties of Noetherian rings.

Ideals: Polynomials and Definitions

Explores ideals in K[X], including PGCD, uniqueness, coprimality, and theorems of Bézout and Gauss.

Polynomials on a Field: Basics and Operations

Introduces the basics of polynomials on a field, focusing on definitions, operations, and properties.

Properties of Euclidean Domains

Covers the properties of Euclidean domains and irreducible elements in polynomial rings.

Dimension Theory of Rings

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

Ideals and PPCM

Covers the concept of ideals in polynomial rings and their properties.

Ring Constructions: Structure Theorems

Explores operations on ideals and structure theorems in commutative rings.

Polynomials: Rings and Operations

Covers the basics of polynomials, focusing on rings, operations, and properties.

Ring Operations: Ideals and Classes

Covers the operations in rings, ideals, classes, and quotient rings.

Principal Ideal Domains: Structure and Homomorphisms

Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.

Field Properties: Irreducibility and Units

Covers the properties of fields, including irreducibility and units in polynomials.

Rings and Fields: Principal Ideals and Ring Homomorphisms

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

Polynomials: Definitions and Operations

Covers the definition and operations of polynomials, including addition and multiplication, degree, coefficients, and their role in algebraic systems.

Ring Structure: Polynomials and Coefficients

Covers the ring structure, focusing on polynomials and coefficients, including associativity, distributivity, and the product of rings.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Congruence Relations in Rings

Explores congruence relations in rings, principal ideals, ring homomorphisms, and the characteristic of rings.

Ideals in Commutative Rings

Covers the concept of ideals in commutative rings and their role in ring homomorphisms.