MATH-310: Algebra

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Lectures in this course (38)

Field Properties: Irreducibility and Units

Covers the properties of fields, including irreducibility and units in polynomials.

Irreducible Polynomials and Finite Fields

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

Polynomial Rings and Irreducibility Criteria

Covers polynomial rings, irreducibility criteria, and algebraic structures in fields.

Algebra: Practice Exam Solutions

Covers the solution of a practice exam in Algebra, focusing on finding greatest common divisors of polynomials and exploring group properties.

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Integers: Sets, Maps, and Principles

Introduces sets, maps, divisors, prime numbers, and arithmetic principles related to integers.

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.

Dihedral Group: Symmetries and Cosets

Explores the symmetries of a regular n-gon, normal subgroups, cosets, and Lagrange's theorem.

Applications of Lagrange's Theorem

Explores Lagrange's theorem applications in group theory and arithmetic, focusing on subgroups, cosets, quotient groups, and homomorphisms.

Symmetric Group and Alternating Group

Explores the Symmetric group, cycle notation, multiplication, the sign of a permutation, and the Alternating group.

Orbit-Stabilizer Theorem: Conjugacy Classes and Direct Products

Covers the Orbit-Stabilizer theorem, conjugacy classes, and direct products in finite groups.

Classification of Finite Abelian Groups

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

Division Rings and Ideals

Covers division rings, fields, and ideals in commutative rings with examples in Z and quaternions.

Principal Ideal Domains: Structure and Homomorphisms

Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.

Chinese Remainder Theorem and Polynomial Rings

Covers the Chinese remainder theorem, polynomial rings, and Euclidean domains among other topics.

Properties of Euclidean Domains

Explores the properties of Euclidean domains, including gcd, lcm, and the Chinese remainder theorem for polynomial rings.

Irreducible Polynomials and Finite Fields

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

Finite Fields and Group Theory

Explores solutions of the 2018 exam, finite fields, group theory, congruences, and polynomial irreducibility in Q[X].

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