Legendre and Jacobi Symbols: RSA Cryptography

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Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Groups: Definitions, Properties, and Homomorphisms

Introduces the basic concepts of groups, including definitions, properties, and homomorphisms, with a focus on subgroup properties and normal subgroups.

Group Homomorphisms

Explores group homomorphisms, isomorphisms, and generators in abstract algebra.

Diffie-Hellman and ElGamal Cryptosystems

Explores Diffie-Hellman key exchange, ElGamal cryptosystem, and their security applications in cryptography.

Classification of Extensions

Explores the classification of extensions in group theory, emphasizing split extensions and semi-direct products.

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Chinese Remainder Theorem: Rings and Fields

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

First Isomorphism Theorem

Covers the first isomorphism theorem in group theory and its applications.

Algebraic Structures: Groups and Rings

Covers groups, rings, number theory, atomic bonds, and materials structure, setting the foundation for further exploration.

Prime Numbers: Finding and Testing

Covers the definition of a function to determine if a given number is prime.