Modern Algebra: Lectures

• Lecture 1: Professor Ralph Cohen of Stanford University introduces and gives an overview of EPGY's Modern Algebra course.
• Lecture 2: Definition of a group, some elementary properties and an example.
• Lecture 3: The example continued: proof of the fact that a set of symmetries is a group.
• Lecture 4: Properties of the group operations relating to the identity element and the inverse element; subgroups.
• Lecture 5: Abelian groups and an example: the cyclic groups.
• Lecture 6: An example: the dihedral groups.
• Lecture 7: Morphisms of groups.
• Lecture 8: Permutations; the symmetry group on n letters.
• Lecture 9: Transpositions; even and odd permutations; cycle decomposition; the alternating group.
• Lecture 10: Cayley's Representation Theorem.
• Lecture 11: Equivalence relations: symmetry, transitivity and reflexivity; equivalence classes.
• Lecture 12: Equivalence relations applied to groups: cosets and Lagrange's Theorem.
• Lecture 13: Group structure on the equivalence classes of a subgroup: normal subgroups and quotient groups.
• Lecture 14: A partial converse of Lagrange's Theorem and the fact that the alternating group on five letters contains no proper normal subgroups.
• Lecture 15: Back to morphisms: kernel and image of a morphism; the morphism theorem.
• Lecture 16: Applications of the morphism theorem.
• Lecture 17: Constructing new groups out of old groups: the direct product.
• Lecture 18: Classification of all groups of order less than or equal to seven.
• Lecture 19: Key application of group theory: group acting on sets as permutations.
• Lecture 20: An example: the group of Euclidean motions acting on points in space.
• Lecture 21: An example: the general linear group of matrices and its classical subgroups.
• Lecture 22: Matrix groups continued: two dimensional matrix groups; rotations and reflections.
• Lecture 23: The rotation groups of the regular solids in three dimensional space.
• Lecture 24: The crystallographic groups: finite order groups of rotations of three dimensional space.
• Lecture 25: Fixed points and orbits of group actions; the Polya-Burnside method for studying symmetry groups.
• Lecture 26: The Polya-Burnside method, continued.
• Lecture 27: Introduction to Rings: definitions, elementary properties and easy examples.
• Lecture 28: Modular arithmetic. Generalizations of familiar rings: integral domains and fields.
• Lecture 29: Subrings; morphisms of rings.
• Lecture 30: Creating new rings from old: the direct product of rings and some examples.
• Lecture 31: The ring of polynomials with coefficients in a given ring.
• Lecture 32: Adding inverses to a ring: the field of fractions over a given ring.
• Lecture 33: More on polynomial rings: division of polynomials; Euclidean domains.
• Lecture 34: A generalization of the division of polynomials: the Euclidean Algorithm. Greatest common divisor.
• Lecture 35: Factorization of elements in a ring; irreducibility; Unique Factorization Domains.
• Lecture 36: Factoring polynomials; the fundamental theorem of algebra.
• Lecture 37: Techniques for factoring polynomials: the rational roots test and Eisenstein's criterion.
• Lecture 38: (Course survey.)
• Lecture 39: Ideals and quotient rings.
• Lecture 40: Quotients of polynomial rings; factorization and irreducibility revisited.
• Lecture 41: More on quotient rings: the morphism theorem for rings.
• Lecture 42: Fields, subfields and extension fields. Certain quotients of polynomial rings as fields.
• Lecture 43: Algebraic and transcendental numbers.
• Lecture 44: The rational, real and complex fields in the context of field extension theory.
• Lecture 45: Finite fields; characteristic of a field; Galois fields.
• Lecture 46: Primitive elements in Galois fields; the group of units in a finite field.
• Lecture 47: Ruler-and-compass constructions and field extensions; the resolution of three famous geometric problems that were unsolved by the early Greeks.
• Lecture 48: More on constructible numbers: a test for constructibility.
• Lecture 49: The impossibility of duplicating a cube or trisecting an angle.
• Lecture 50: The impossibility of squaring the circle. Constructing regular polygons.