Math 333. Abstract Algebra I


Homework




Lectures (The "a"-versions of lecture notes are for 10:30 classes, "b" for 11:30.)

  • Aug.29: Going over course syllabus. Course overview. Notes.

  • Aug.31: Binary operations. Definition and first examples of groups. Some first consequences of group axioms. Worksheet 1(solutions). Notes.

  • Sep.02: Inverses of products. Symmetric groups and dihedral groups. Worksheet 2(solutions). Notes a . Notes b.

  • Sep.07: Notations for and computations in dihedral groups. Cycle notation for symmetric groups. Worksheet 3 (solutions). Notes a. Notes b.

  • Sep.09: More on additive vs multiplicative notation. Working with exponentials. Definition, examples, and recognition criteria for subgroups. Notes a. Notes b.

  • Sep.12: Definition of a subgroup generated by a subset X of a group (as an intersection), and some alternative descriptions of such subgroups. Notes a. Notes b. Worksheet 4 (solutions).

  • Sep.14: Subgroups of (Z,+) are all cyclic. Definitions and first properties of homomorphism and isomorphisms. Notes a. Notes b.

  • Sep.16: All infinite cyclic groups are isomorphic to Z. Ubiquity of homomorphisms. Definition of cosets. Notes a. Notes b.

  • Sep.19. Applications of cosets: Lagrange's theorem, first steps towards quotient groups. Notes a. Notes b. Worksheet 5 (solutions).

  • Sep.21: More on well-definedness. Normal subgroups. Notes a. Notes b. Worksheet 6 (solutions).

  • Sep.23: Normality and quotient groups. Kernels of homomorphisms. The first isomorphism theorem. Notes a. Notes b.

  • Sep.26: Proof and first applications of the first isomorphism theorem. Notes a. Notes b.

  • Sep.28: More applications of the first isomorphism. Worksheet on the second isomorphism theorem. Notes a. Notes b. Worksheet 7 (see Sept 30 notes for solutions).

  • Sep.30: Proof of the second isomorphism theorem. Midterm Q&A. Notes a. Notes b.

  • Oct.03: Definition and first examples of rings. Notes a. Notes b. Worksheet 8 (solutions).

  • Oct.05: Basic properties of rings. Cancellation of multiplication in rings. Subring tests. Notes a. Notes b.

  • Oct.07: Definition, examples, and observations for ring homomorphisms and ideals. Notes a. Notes b.

  • Oct.17: Review on rings and ideals (after the fall break). More examples and nonexamples of ideals. Notes a. Notes b. Worksheet 9 (solutions).

  • Oct.19: From two-sided ideals to quotient rings. ("Two-sided ideals are to rings what normal subgroups are to groups.") The first isomorphism theorem for rings. Hints for HW5P6. Notes a. Notes b. Worksheet 10 (solutions).

  • Oct.21: The second and third isomorphism theorems. The correspondence theorem. Notes a. Notes b.

  • Oct.24: An application of the correspondence theorem. Prime ideals and their connection to prime numbers. Notes a. Notes b.

  • Oct.26: Discussion about mid-semester evaluations. Prime ideals and integral domains. Fields. Notes a. Notes b. Worksheet 11 (solutions).

  • Oct.28: More discussion on quotient groups and rings. Closer examination of the complex field. Notes a. Notes b. Worksheet 12 (solutions).

  • Oct.31: Complex conjugation. Norm of complex numbers. Maximal ideals and fields. Notes a. Notes b.

  • Nov.02: Finishing the main theorem on maximal ideals and fields. Hints on HW7. Notes a. Notes b.

  • Nov.04: Polynomial rings. How to and how not to think about associativity of polynomial multiplication. Notes a. Notes b. Worksheet 13 (solutions).

  • Nov.07: Polynomial rings over integral domains. Evaluation homomorphisms. Polynomial division. Notes a. Notes b.

  • Nov.11: The division algorithm for polynomials and its applications. Roots of polynomials. Notes a. Notes b. Worksheet 14 (solutions).

  • Nov.14: Roots vs. linear factors of polynomials. Number of roots vs degree of polynomials. Multivariate polynomial rings. Notes a. Notes b.

  • Nov.16: F[x,y] is not a PID. Irreducible polynomials: definitions, connection to maximal ideals, and irreducibility tests for polynomials of low degree. Notes a. Notes b. Worksheet 15 (see the lecture notes of Nov.18 for solutions).

  • Nov.18: The complex field via \R[x]. Gauss' Lemma. Notes a. Notes b.

  • Nov.21: Applications of Gauss' Lemma. Eisenstein's criterion. Cyclotomic polynomials. Notes a. Notes b.

  • Nov.28: Factorization in general commutative rings. Irreducible vs prime elements. UFDs. Notes a. Notes b.

  • Nov.30: Ascending chains in PIDs stablize. Proof that PIDs are UFDs. Notes a. Notes b.

  • Dec.02: Extension fields and roots of polynomials. Notes a. Notes b. Worksheet 16 (solutions).

  • Dec.05: Splitting fields. Numerical constraints on finite fields. Note a. Notes b.

  • Dec.07: Orders of finite fields. Classification of finite fields. Notes a. Notes b.

  • Dec.09. Finishing the proof of the classification theorem for finite fields. Review questions. Notes a. Notes b.