Math 225 Modern Algebra

Fall 2017
Professor D.Joyce
Department of Mathematics and Computer Science
Clark University

Emmy Noether

Emmy Noether (1882–1935)
Zbrush sculpture
Date 2014, by Doris Fiebig

Instructor: David Joyce
BP 322. Phone: 793-7421
Home page: http://aleph0.clarku.edu/~djoyce/
Clark University Academic Catalog, Math 225
In the 19th century, Kummer introduced "ideal numbers" to salvage unique factorization of integers into primes (which breaks down in some rings of algebraic integers). This course discusses unique factorization and the modern theory of rings and their ideals, emphasizing Euclidean domains. Other algebraic structures (groups, fields) also are introduced. Required for all mathematics majors. Prerequisite: Math 130 Linear Algebra.
Course goals.
- To provide students with a good understanding of the theory of modern algebra as described in the syllabus.
- To help students develop the ability to prove theorems and solve problems.
- To introduce students to some of the basic methods of modern algebra.
- To develop abstract and critical reasoning by studying logical proofs and the axiomatic method as applied to modern algebra.
- To make connections between modern algebra and other branches of mathematics, and to see some of the history of the subject.
Text: The text is online at algebra.pdf.
Syllabus.
- Background on logic and set theory
  - Writing Proofs: structures of theorems and proofs, logical symbols, and well-written proofs
  - Kinds of proofs: synthetic and analytic proofs
  - A little bit about sets
- Algebraic structures: introduction to fields, rings and groups
  - Operations, commutativity, associativity, identity elements, inverse elements, distributivity
  - Fields: definition and examples. The fields of real numbers R, of rational numbers Q, and of complex numbers C
  - Rings: definition and examples. Commutative and noncommutative rings The ring of integers Z, polynomial rings R[x], matrix rings M_n(R), the ring of integers modulo n Z_n
  - Groups: definition and examples. Abelian and nonabelian groups The multiplicative group of invertible elements in a ring, groups of symmetries of geometric figures
  - Isomorphisms and homomorphisms of rings and groups
  - Polynomials
- A little number theory
  - Mathematical induction, the well-ordering principle
  - Divisibility, greatest common divisor, prime numbers
  - The Euclidean algorithm and the extended Euclidean algorithm
  - The unique factorization theorem, a.k.a, the fundamental theorem of arithmetic
- Fields
  - Formal definition. Subfields.
  - F(x) rational functions with coefficients in a given field
  - Properties
  - Algebraic fields
  - Equivalence relations and congruence
  - Congruence modulo n
  - Equivalence relations, equivalence classes, partitions, quotient sets
  - Cyclic rings Z_n
  - Prime fields Z_p and other Galois fields, characteristic of a field F
  - Field extensions, algebraic fields
  - Ordered fields, complete ordered fields, and R
  - Division rings, quaternions H
- Rings
  - Formal definition. Commutative rings. Rings with 1. Subrings
  - Properties
  - Examples besides fields. Z, Z_n, matrix rings
  - R[x] polynomials with coefficients in a given commutative ring R
  - Products of rings
  - Chinese remainder theorem
  - Boolean rings
  - The category of rings, ring homomorphisms, isomorphisms, monomorphisms, etc.
- Integral domains
  - Definition. The cancellation law
  - Subrings of integral domains and fields. The Gaussian integers
  - Finite integral domains are fields
  - The field of fractions of an integral domain
- Ideals and quotient rings
  - The kernel of a ring homomorphism
  - Definition of ideal, examples
  - The ideal generated by a subset
  - Construction of the quotient ring from an ideal
  - Prime and maximal ideals
  - Zorn's lemma, Krull's theorem
- Special integral domains: UFDs, PIDs, and EDs
  - Divisibility, irreducible elements, prime elements in an integral domain
  - Definition and examples of Unique Factorization Domains (UFDs)
  - Definition and examples of Principal Ideal Domains (PIDs)
  - PIDs are UFDs
  - Definition and examples of Euclidean Domains (EDs)
  - The division algorithm for polynomials
  - Gaussian integers
  - EDs are PIDs
- Polynomial rings
  - Basic properties of polynomials
  - Complex polynomials and the fundamental theorem of algebra
  - Algebraically closed fields
  - Real polynomials
  - Polynomials with integer and rational coefficients, criteria for irreducibly
  - Gauss's lemma, Eisenstein's criterion
  - Polynomials with coefficients in a UFD, polynomial rings in several variables
- Introduction to groups
  - Definition and examples
  - Basic properties of groups and subgroups, cyclic groups
  - Cosets, Lagrange's theorem
  - Permutations, symmetric groups, alternating groups, dihedral groups
  - Cayley's theorem and Cayley graphs
  - The category of groups
  - Conjugacy classes and quandles
  - Kernels, normal subgroups, quotient groups, isomorphism theorems
  - Matrix rings and linear groups: general, special, orthogonal, unitary, projective
  - Simple groups, the Jordan-Holder theorem
  - Abelian groups, the category of Abelian groups, finite Abelian groups
Homework, quizzes, tests, and course grade. There will be weekly homework, a few short quizzes, one or two in-class midterm tests, and a final examination during finals week.
Come to class every day. Expect to put in several hours of work outside of class each week.
Class notes
- Tuesday, Aug 29. Welcome and overview of the course; fields, rings, groups in section 1.2; N, Z, Q, R, C
  The first homework is due Tuesday, Sep 5. Exercises 1 and 2 from the text.
- Friday, Sep 1. More on groups. A Group of order 6; Z_n
- Tuesday, Sep 5. Review of logic and proofs; some set theory in appendix A.2; Isomorphisms of rings and groups in section 1.3.
  Homework due, exercises 1 and 2. Answers
  Due next Tuesday: exercises 3 through 7.
- Friday, Sep 8. Homomorphisms of rings and groups, monomorphisms and epimorphisms, endomorphisms and automorphisms; a little number theory in section 1.4, mathematical induction, well-ordering principle, divisibility
  Due next Friday: exercises 7 through 13.
- Tuesday, Sep 13. Prime numbers, infinity of primes; Euler's totient function; greatest common divisors, the Euclidean algorithm and the extended Euclidean algorithm; least common multiples
  Homework due, exercises 3 through 6. Answers
- Friday, Sep 15. The unique factorization theorem a.k.a the fundamental theorem of arithmetic; division algorithm for polynomials; review of complex numbers
  Homework due, exercises 7 through 13. Answers
  Due next Friday, exercises 14 and 17 through 26.
- Tuesday, Sep 19. Formal definition of fields, subfields, Q, vector spaces, cyclic rings Z_n
- Friday, Sep 22. Finite fields Z_p and cyclic rings Z_n, characteristic of a field, prime fields, the Frobenius endomorphism; algebraic numbers, algebraic integers, algebraic field extensions, transcendental numbers, transcendental field extensions.
  Homework due, exercises 14 and 17 through 26. Answers
  Due next Friday, exercises 27 and 28.
- Tuesday, Sep 26. The field of complex numbers C, complex conjugation, matrix representation of C; quadratic extensions; ordered fields
- Friday, Sep 29. Archimedean fields, surreal numbers, complete ordered fields, the field of real numbers
  Homework due, exercises 27 and 28. Answers
  Due next Friday, exercises 29 through 32.
  Select date for first midterm to cover through the end of chapter 2.
- Tuesday, Oct 3. Skew fields, Quaternions H
- Friday, Oct 6. Begin chapter 3 on rings.
  Tuesday, Oct 10. Fall break, no class.
- Friday, Oct 13. Review for test. Integral domains.
- Tuesday, Oct 17. First midterm. Covers chapters 1 and 2. Answers.
- Friday, Oct 20. Gaussian integers; Chinese remainder theorem; intro Boolean rings.
  Due next Friday, exercises 33, 34, and 35.
- Tuesday, Oct 24. Boolean rings.
- Friday, Oct 27. More on Boolean rings; definition of a category.
  Homework due, exercises 33, 34, and 35. Discussed in class.
- Tuesday, Oct 31. Categories of sets, rings, and groups; products in categories.
  Due next Tuesday. Exercises on Chinese remainder theorem and on initial & final objects.
- Friday, Nov 3. Initial and final objects in a category; monomorphisms and epimorphisms; kernels of ring homomorphisms.
- Tuesday, Nov 7. Ideals in a ring, principal ideals; quotient rings; prime ideals.
  Homework due, exercises on Chinese remainder theorem and on initial & final objects. Answers
- Friday, Nov 10. Class cancelled.
- Tuesday, Nov 14. Prime and maximal ideals; axiom of choice, Krull's theorem; divisibility in an integral domain.
- Friday, Nov 17. Review of rings.
  Homework due: xercises 38 through 41. Answers.
- Tuesday, Nov 21. Second midterm. Covers 3.1 through 3.6. Answers.
- Friday, Nov 24. Thanksgiving holiday, no class.
- Tuesday, Nov 28. Return second midterm.
  Unique factorization domains (UFD's). Principal ideal domains (PID's).
  Due next Tuesday, exercises 42 and 43.
- Friday, Dec 1. PID's and Euclidean Domains.
- Tuesday, Dec 5. Real and complex polynomial rings, the Fundamental Theorem of Algebra, rational and integral polynomial rings, Gauss's lemma, Eisenstein's criterion
  Homework due, exercises 42 and 43. Answers
  Due Friday, exercises 44 and 45.
- Friday, Dec 8. Last day of regular class meetings.
  Homework due, exercises 44 and 45.
- Monday, Dec 18. 8:00-10:00 a.m. Room BP316. Final exam.
Related courses and resources
- Writing Proofs: structures of theorems and proofs, synthetic and analytic proofs, logical symbols, and well-written proofs
- A little bit about sets
- Notes on LaTeX
- Notes on Matlab
- Dave's short course on complex numbers at http://www.clarku.edu/~djoyce/complex/
- Math 126 Number Theory
- Math 130 Linear Algebra
- Notes on Richard Dedekind's Was sind und was sollen die Zahlen?
- Wallpaper groups
- An Algebraic Approach to Symmetry with Applications to Knot Theory
Past tests
Pages on the web that you may find interesting.
- MathArchives http://archives.math.utk.edu/ links for Abstract algebra

http://aleph0.clarku.edu/~djoyce/ma225/

Math 225 Modern Algebra

Fall 2017 Professor D.Joyce Department of Mathematics and Computer Science Clark University

Fall 2017
Professor D.Joyce
Department of Mathematics and Computer Science
Clark University