Proof of Lagrange Theorem

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Ring of Polynomials: Coefficients and Multiplication

Covers the ring of polynomials, focusing on coefficients and multiplication.

Classification of Finite Abelian Groups

Covers the classification theorem for finite abelian groups and introduces rings, including zero divisors and domains.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Irreducible Factors and Noetherian Rings

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

Algebraic Structures of Morphism Spaces

Explores algebraic structures of morphism spaces, including stability properties and injective ring homomorphisms.

Fourier Expansion and Quaternion Algebra

Discusses Fourier expansion, quaternion algebra, and Whittaker model in automorphic forms.

Integrity in Algebra

Explores the properties of commutative rings and integral domains in algebra.

Orthogonality and Least Squares Methods

Explores orthogonality, norms, and distances in vector spaces for solving linear systems.

Polynomials: Definition and Operations

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

Cyclotomic Extensions: Norms, Ideals, and Primes

Explores cyclotomic extensions, prime numbers, and ideal norms in number theory.