Florida Atlantic University • Fall 2019 • Markus Schmidmeier

# Introductory Abstract Algebra II

Welcome to my course! We meet Wednesday and Friday 2:00 - 3:20 p.m. in BU 403. Course numbers are MAS 5312 for graduate credit or MAS 4306 for undergraduate credit. Prerequisite for this course is Introductory Abstract Algebra I.

Parallel to this course, there are example classes on Monday 2:00 - 3:20 p.m. in SE 215, see below.

Textbook

I. N. Herstein, Topics in Algebra, 2nd edition, Wiley 1975, ISBN-10: 0471010901, ISBN-13: 978-0471010906.

Topics

Fields play a central role in algebra and its applications. Results about them have impact on the theory of numbers. Field theory encompasses the theory of equations which treats questions about the roots of polynomials. And the study of certain field extensions yields the answers to problems left open in greek mathematics about the doubling of the cube, the trisection of the angle, the constructability of regular polygons, and last but not least, the transcendence of π.

Key topic of this course are the beautiful ideas, due to the brilliant French mathematician Evariste Galois, which have served as a guiding inspiration for algebra as it is today (Herstein, Chapter 5.6).

In this course, we will resume our study of rings in Chapter 3.6ff (Euclidean rings). Prime examples are the ring of integers and the polynomial ring. Nomen est omen: It is the Euclidean Algorithm which provides access to ring theoretic properties like principal ideal ring, or unique factorization domain.

Last semester we have seen that finite abelian groups are direct sums of cyclic groups. But abelian groups are just modules over the integers! We generalize the result just mentioned to finitely generated modules over a Euclidean ring. (Chapter 4.5)

Key topic in this course is the theory of field extensions a.k.a. Galois Theory (Chapter 5), with its application to constructions with ruler and compass.

As time permits, we will study finite fields (Chapter 7.1).

Example Classes

Every Monday, 2:00 - 3:20 p.m. in SE 215, Ms. Alexandra Milbrand will be available for discussion of topics covered in class.

Note that Ms. Milbrand is not expected to solve homework problems for you, and she is not permitted to give updates on what will be on the quiz.

Please take part in the example classes to review what we have covered in class, in particular previous homework and quizzes.

Credit

Homework:  There will be weekly

homework assignments,

I will grade homework every other week. The 6 best count for 20% of the grade.

Quizzes:  Every second week, we will have a quiz. The 6 best quizzes count for 20% of the grade.

Presentations:  Two presentations, one before February 15, and both before April 5, each solving a problem and taking at most 10 minutes, will together count for 20% of the grade.

Final Exam:   The final exam on Wednesday, May 1, 1:15 - 3:45 p.m. in our classroom is comprehensive and counts for 40% of the grade.

Further Information

For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale and Financial Assistance Opportunities please visit Infos for all my courses.

Contact Me

Office hours:   WF 3:30 - 5:00 p.m. in SE 272 or by appointment

E-mail:   markus@math.fau.edu,    Phone (office):  561-297-0275.