Lectures related to Polynomial ring

Minimal Polynomials: Uniqueness and Division

Explores the uniqueness of minimal polynomials and the division algorithm for polynomials.

Division Polynomials: Theorems and Applications

Explores division polynomials, theorems, spectral values, and minimal polynomials in endomorphisms and vector spaces.

Explores eigenvalues and minimal polynomial, emphasizing their importance in linear algebra.

Explores system equivalence, state-space representation, transfer functions, and Euclidean rings, emphasizing unimodular matrices and their properties.

Algebraic Curves: Normalization

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Decomposition of Linear Operators

Covers the decomposition of linear operators and properties of eigenspaces.

Irreducible Polynomials and Finite Fields

Explores irreducible polynomials, finite fields, cyclic unit groups, and field construction.

Irreducible Polynomials and Finite Fields

Explores irreducible polynomials, finite fields, and the construction of unique finite fields from irreducible polynomials.

Properties of Euclidean Domains

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

Commutativity in Rings

Covers commutativity in rings, evaluation options, and formal linear combinations.

Chinese Remainder Theorem: Rings and Fields

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

Introduction to Finite Fields

Covers the basics of finite fields, including arithmetic operations and properties.

Polynomials on a Field

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

Eigenvalues and Eigenvectors

Covers eigenvalues and eigenvectors, explaining their importance in linear algebra.

Stein Algorithm: Polynomial Identity Testing

Explores the Stein algorithm for polynomial identity testing and the minimization of a cut problem.

Polynomial Methods: GCD Calculation Summary

Covers the calculation of the greatest common divisor using polynomial methods and the Euclidean algorithm.

Error-Correcting Codes: Hamming Distance and Polynomial Multiplication

Covers error-correcting codes, Hamming distance, and polynomial multiplication in algebraic fields.

Polynomial Factorization and Decomposition

Covers polynomial factorization, irreducible polynomials, ideal decomposition, and the theorem of Bézout.

Polynomials on a Field: Properties and Applications

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

Finite Fields: Construction and Properties

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