MATH 403/503 Intermediate Abstract Algebra

Spring 2024

Instructor

Grader

Time and Place

Topics Covered

Properties of groups and rings, subgroups, ideals, and quotients, homomorphisms, structure theory for finite groups. PIDs, UFDs, and Euclidean Domains. Field extensions and finite fields. Selected applications.

Texts and Resources

Exams and Grading

Your homework should be neat and stapled. Problems should be listed in order and proofs given in complete sentences with correct notation. Use of LaTeX is encouraged but not required. For each homework question you must:

Each problem will be graded on the following 4 point scale:

4=Very Good - Correct mathematics that is carefully thought out and thoroughly explained with complete sentences.

3=Good - Correct mathematics with some minor gaps in explanation.

2=Basic - Partial solution with errors in understanding or computation.

1=Emerging - Work that has some merit but also has significant shortcomings in the mathematics or explanation.

0=No Credit - No work submitted or no serious attempt.

Homework and Practice Problems

Homework sets will be posted here. They are due most Wednesdays. First homework is due January 24.

  1. (Due Jan 24) Homework 1, LaTeX file

  2. (Due Jan 31) Homework 2, LaTeX file

  3. (Due Feb 7) Homework 3, LaTeX file

  4. (Due Feb 14) Homework 4, LaTeX file

    (Need not hand in) Practice Problems for Exam 1

  5. (Due March 6) Homework 5, LaTeX file

  6. (Due March 20) Homework 6, LaTeX file

  7. (Due March 27) Homework 7, LaTeX file

    (Need not hand in) Practice Problems for Exam 2

  8. (Due April 10) Homework 8, LaTeX file

  9. (Due April 17) Homework 9, LaTeX file

  10. (Due April 24) Homework 10, LaTeX file

    (Need not hand in) Practice Problems for Final Exam with Some Solutions

Lecture Summary

A short summary of each lecture will be posted here.

[J] refers to Judson (July 8, 2022 version).

[H] refers to Hungerford, Abstract Algebra – An Introduction, 3rd Edition.

  1. Lecture 1 Monoids, units, homomorphisms, kernel.
  2. Lecture 2 Congruence relations on monoids. Quotient monoids. Congruence relations on groups. Normal subgroups [J §10.1], [H §8.2].
  3. Lecture 3 Quotient groups [J, §10.1], [H, §8.3].
  4. Lecture 4 The First Isomorphism Theorem [J §11.2], [H §8.4].
  5. Lecture 5 Symmetric group [J §5.1], [H §7.5]. Applying the Isomorphism Theorem, [J §11.2], [H §8.4].
  6. Lecture 6 Conjugacy classes in the symmetric group \(S_n\) versus partitions of \(n\). Example: \(N\trianglelefteq S_4\) with \(S_4/N\cong S_3\).
  7. Lecture 7 Second Isomorphism Theorem [J §11.2], [H §8.4 Problem 40(!)].
  8. Lecture 8 Correspondence Theorem and the Third Isomorphism Theorem [J §11.2], [H §8.4].
  9. Lecture 9 Group actions [J §14.1].
  10. Lecture 10 Orbit-Stabilizer Theorem. Class Equation. Nontrivial \(p\)–groups have nontrivial center. [J §14.1, §14.2], [H p. 305–306; in §9.4].
  11. Lecture 11 Burnside’s Counting Formula [J §14.3].
  12. Lecture 12 Sylow’s Theorem [J §15.1], [H §9.3].
  13. Lecture 13 Proof of Sylow’s Theorem [J §15.1], [H §9.3], [Dummit and Foote, §4.5].
  14. Lecture 14 Sylow examples [J §15.2], [H §9.3]. Finitely Generated Abelian Groups, [J §13.1], [H §9.2].
  15. (Review)
  16. (Review)
  17. (Exam #1)
  18. Lecture 18 Rings and ring homomorphisms. Matrix rings, polynomial rings, endomorphism rings, real quaternion ring. Units, division rings, fields. The ring \(\mathbb{F}[x]/(p(x))\). [J §16.1, §17.1] and [H §3.1].
  19. Lecture 19 Basic properties of rings. Subrings. Zero-divisors and integral domains. [J §16.1, §16.2].
  20. Lecture 20 Finite integral domains are fields. Ideals. Kernels. Principal ideals. [J §16.2, §16.3].
  21. Lecture 21 Quotient rings and First Isomorphism Theorem for Rings [J §16.3]; Prime Ideals and Integral Domains [J §16.4].
  22. Lecture 22 Evaluation homomorphism, maximal ideals and fields [J §16.3, §16.4].
  23. Lecture 23 Polynomials and Irreducibility criteria [J §17].
  24. Lecture 24 Fraction fields [J §18.1]
  25. Lecture 25 Factorization, PIDs, UFDs [J §18.2]
  26. Lecture 26 Every PID is a UFD. Euclidean domains (EDs). Every ED is a PID [J §18.2].
  27. Lecture 27 Factorization in \(D[x]\) [J §18.2].
  28. (Review)
  29. (Review)
  30. Exam #2
  31. Lecture 31 Field extensions, Fundamental Theorem of Field Theory, algebraic/transcendental elements [J §21.1].
  32. Lecture 32 Minimal polynomial, degree of a field extension, degree formula [J §21.1].
  33. Lecture 33 Proof of Dimension Formula, finite extensions are algebraic [J §21.1].
  34. Lecture 34 Splitting fields [J §21.2].
  35. Lecture 35 Algebraic closure [J §21.1].
  36. Lecture 36 Constructions with straightedge and compass [J §21.3].
  37. Lecture 37 \(\mathbb{R}_c\) is a field [J §21.3].
  38. Lecture 38 Elements of \(\mathbb{R}_c\) have degree power of \(2\) [J §21.3].
  39. Lecture 39 Finite fields, separability [J §22.1].
  40. Lecture 40 The Galois group of a field extension [J §23.1].
  41. Lecture 41 The Fundamental Theorem of Galois Theory [J §23.2].
  42. Lecture 42 Solvability by radicals [J §23.3].
  43. (Review)
  44. (Review)

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