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
  5. (Need not hand in) Practice Problems for Exam 1
  6. (Due March 6) Homework 5, LaTeX file

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].

Statement on Free Expression

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.


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