Lectures related to Euclidean Division and Simple Elements

Decimal Expansion: Division and Periodicity

Delves into decimal expansion of rational numbers through Euclidean division, emphasizing periodicity and illustrative examples.

Minimal Polynomials: Uniqueness and Division

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

Euclidean Division: Uniqueness and Remainder

Explores Euclidean division for polynomials, emphasizing uniqueness of quotient and remainder.

Algebraic Geometry: Rings and Bodies

Explores algebraic geometry, focusing on rings, bodies, quotient rings, and irreducible polynomials.

Decimal Writing: Fractions and Division

Explains the conversion between decimal and fractional writing, including division with remainder and a trick for periodic numbers.

Rational Numbers: Amplification and Simplification

Covers rational numbers, amplifying and simplifying fractions, and calculating values.

Division and Multiplication in Rational Numbers

Explores division and multiplication in rational numbers, emphasizing the concept of multiplying by the inverse.

Division Polynomials: Theorems and Applications

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

Multiplication in Rational Numbers

Covers the extension of addition and multiplication operations in rational numbers.

Polynomial Factorization: Field Approach

Covers the factorization of polynomials over a field, including division with remainder and common divisors.

System Equivalence

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

Polynomial Methods: GCD Calculation Summary

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

Polynomial Factorization over Finite Fields

Introduces polynomial factorization over finite fields and efficient computation of greatest common divisors of polynomials.

Chinese Remainder Theorem and Euclidean Domains

Explores the Chinese remainder theorem, systems of congruences, and Euclidean domains in integer numbers and polynomial rings.

Polynomial Division: Terms and Factors

Covers polynomial division, terms, factors, and integration methods.

Rational Numbers: Construction and Properties

Covers the construction and properties of rational numbers, including fractions and equivalence, with proofs on commutativity.

Integration of Rational Functions

Covers the integration of rational functions, including the decomposition into partial fractions and the use of complex zeros.

Properties of Euclidean Domains

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

Decomposition of Fractions: Simple and Corollary Proof

Covers the decomposition of fractions into simple terms and provides a corollary proof related to the convergence of integrals.

Polynomial Factorization over a Field: Eigenvalues

Explores polynomial factorization over a field, emphasizing eigenvalues and irreducible components.