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Hurwitz's theorem (composition algebras)
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Representation Theory: Algebras and Homomorphisms
Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.
Analytic Manifolds and Berkovich Spaces
Introduces analytic manifolds, Berkovich spaces, and multiplicative semi-norms in K-algebras.
Analysis IV: Convolution and Hilbert Structure
Explores convolution, uniform continuity, Hilbert structure, and Lebesgue measure in analysis.
Decimal Expansion: Division and Periodicity
Delves into decimal expansion of rational numbers through Euclidean division, emphasizing periodicity and illustrative examples.
Fermat's Theorem: Sums of Squares
Explores Fermat's Theorem, factorization of integers, properties of Z[i], and Hurwitz quaternions.
Proof of Lagrange Theorem
Covers the proof of Lagrange Theorem and explores quaternions, prime numbers, and equations.
Simple Modules: Schur's Lemma
Covers simple modules, endomorphisms, and Schur's lemma in module theory.
Stochastic Calculus: Lecture 1
Covers the essentials of probability, algebras, and conditional probability, including the Borel o-algebra and Poisson processes.
Number Theory: Foundations and Applications in Cryptography
Introduces number theory and its essential applications in cryptography.
Algebraic Geometry: Rings and Bodies
Explores algebraic geometry, focusing on rings, bodies, quotient rings, and irreducible polynomials.
Polynomial Equations: Solving Methods
Covers various methods for solving polynomial equations through examples.
Algorithms for Big Numbers: Z_n and Orders
Covers algorithms for big numbers, Z_n, and orders in a group, explaining arithmetic operations and cryptographic concepts.
Euclidean Division: Uniqueness and Remainder
Explores Euclidean division for polynomials, emphasizing uniqueness of quotient and remainder.
Polynomial Factorization over Finite Fields
Introduces polynomial factorization over finite fields and efficient computation of greatest common divisors of polynomials.
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