Introductory Set Theory And Abstract Algebra

1. Define and apply set operations and relations.
2. Explain and apply group theory concepts, including subgroups and Lagrange's Theorem.
3. Define and identify integral domains and fields.
4. Construct polynomial rings and apply the division algorithm.
5. Determine irreducibility of polynomials using Eisenstein's criterion.
6. Construct field extensions and identify prime fields.

13 weeks, about 39 hours in total. Yours will differ.

1. Week 1Module 1: Introduction

   • Unit 1: Sets and Functions · 3 hours

     Review definitions of sets, subsets, and set operations.. Practice problems involving unions, intersections, and complements of sets.. Solve problems related to Cartesian products and relations..

2. Week 2Module 1: Introduction

   • Unit 1: Sets and Functions · 3 hours

     Define and identify reflexive, symmetric, and transitive relations.. Determine equivalence classes for given equivalence relations.. Practice problems involving different types of functions: one-to-one, onto, and bijective..

3. Week 3Module 1: Introduction

   • Unit 2: Groups · 3 hours

     Define binary operations and determine if they are commutative or associative.. Identify identity elements and inverses for given binary operations.. Practice problems involving groups, subgroups, and abelian groups..

4. Week 4Module 1: Introduction

   • Unit 2: Groups · 3 hours

     Apply cancellation laws and laws of indices to solve group-related problems.. Study the properties of integers modulo n and solve related problems.. Explore the symmetric group and its properties..

5. Week 5Module 1: Introduction

   • Unit 3: Subgroups · 3 hours

     Define subgroups and check if a subset of a given group is a subgroup.. Explore properties of subgroups, including intersection and union.. Solve problems related to cyclic groups and their generators..

6. Week 6Module 1: Introduction

   • Unit 4: Lagrange's Theorem · 3 hours

     Form left and right cosets of a subgroup.. Partition a group into disjoint cosets of a subgroup.. Apply Lagrange's Theorem to solve problems related to group and subgroup orders..

7. Week 7Module 2: Advanced Topics

   • Unit 1: The Basics · 3 hours

     Define integral domains and check if an algebraic system is an integral domain.. Obtain the characteristic of any ring.. Check whether an algebraic system is a field or not..

8. Week 8Module 2: Advanced Topics

   • Unit 1: The Basics · 3 hours

     Define and identify prime ideals and maximal ideals.. Prove and use simple properties of integral domains and fields.. Construct or identify the field of quotients of an integral domain..

9. Week 9Module 2: Advanced Topics

   • Unit 2: Polynomial Rings · 3 hours

     Identify polynomials over a given ring.. Prove and use the fact that R[x], the set of polynomials over a ring R, is a ring.. Relate certain properties of R[x] to those of R..

10. Week 10Module 2: Advanced Topics

    • Unit 2: Polynomial Rings · 3 hours

      Prove and use the division algorithm for F[x], where F is a field.. Solve problems related to roots of polynomials.. Apply the Remainder Theorem to find remainders of polynomial divisions..

11. Week 11Module 2: Advanced Topics

    • Unit 3: Special Integral Domains · 3 hours

      Check whether a function is a Euclidean valuation or not.. Identify principal ideal domains.. Identify unique factorization domains..

12. Week 12Module 2: Advanced Topics

    • Unit 3: Special Integral Domains · 3 hours

      Obtain the g.c.d of any pair of elements in a unique factorization domain.. Prove and use the relationship between Euclidean domains, principal ideal domains, and unique factorization domains..

13. Week 13Module 2: Advanced Topics

    • Unit 4: Irreducibility and Field Extensions · 3 hours

      Prove and use Eisenstein's criterion for irreducibility in Q[x].. Obtain field extensions of a field F from F[x].. Obtain the prime field of any field..