Lectures related to Polynomials: Theory and Operations

Properties of Euclidean Domains

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

Polynomials on a Field: Properties and Applications

Explores the properties and applications of polynomials on a field, including formal derivation and uniqueness.

Chinese Remainder Theorem: Rings and Fields

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

Polynomials: Roots and Factorization

Explores polynomial roots, factorization, and the Euclidean algorithm in depth.

Polynomials, Division, and Ideals

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

Ideals: Polynomials and Definitions

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

Irreducible Factors and Noetherian Rings

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

Field Properties: Irreducibility and Units

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

Complex Roots and Polynomials

Explores complex roots, polynomials, and factorizations, including roots of unity and the fundamental theorem of algebra.

Properties of Euclidean Domains

Explores the properties of Euclidean domains, including gcd, lcm, and the Chinese remainder theorem for polynomial rings.

Ideals and PPCM

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

Rings and Fields: Principal Ideals and Ring Homomorphisms

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

Finite Fields: Construction and Properties

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

Polynomes: Irreducible Polynomials and Gaussian Lemma

Introduces irreducible polynomials and the Gaussian lemma for polynomial factorization.

Polynomials: Definition and Operations

Covers polynomials, their operations, division theorem, and provides illustrative examples.

Division Euclidienne: Exemples

Explains the Euclidean division of polynomials and demonstrates its application through examples and root-based divisibility.

Polynomials on a Field

Covers the construction of polynomials on a field and the concept of minimal polynomials.

Complex Polynomials and Factorization

Explores complex polynomials, factorization, roots of equations, equilateral triangles, and infinite sums in sequences.

Factorisation: Real Coefficients Examples

Covers the factorization of polynomials with real coefficients in the complex domain, demonstrating how to find complex roots and obtain irreducible factors.

Galois Theory: Dedekind Rings

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