# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 150C: Modern Algebra

Approved: 2003-04-01 (revised 2012-08-01, Vazirani, Kuperberg, and Schwarz)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Algebra by Michael Artin; Addison Wesley; 2nd edition (August 13, 2010); ISBN-10: 0132413779; Price ranges from \$111.00 to \$145.00.
Search by ISBN on Amazon: 978-0132413770
Prerequisites:
MAT 150B.
Suggested Schedule:
 Lecture(s) Sections Comments/Topics Week 1 Definition of a group representation (review some of Chap. 5 of Artin if needed); G-invariant subspaces and irreducible representations. Week 2 Permutation representations and the regular representation; Characters. Week 3 Orthogonality relations; One-dimensional representations; Schur’s Lemma. Week 4 SU(2); Abstract fields; matrix groups and linear algebra over abstract fields (from Chap. 3 of Artin); Adjunction of Elements (Chap. 10). Week 5 Examples of fields; Real quadratic fields (Chap. 11). MIDTERM Week 6 Algebraic and transcendental elements; Field extensions. Week 7 Constructions with ruler and compass; Finite fields (recall we can do linear algebra over finite fields); Function fields. Week 8 Algebraically closed fields; Fundamental theorem of Galois theory. Week 9 Cubic equations; Primitive elements; Symmetric functions. Week 10 Cyclotomic extensions.
The class is based primarily on Chapters 9, 12-14 of Artin’s book.
Learning Goals:

In this course students will:

1. Understand the major theorems and their proofs;
2. Prove theorems in general algebraic settings;
3. Apply general algebraic theorems in specific instances; and
4. Understand the fundamental principles underlying the algebra with which they are familiar.

In particular, by the end of this course, students will develop intuition about the objects of modern algebra, have more finely developed proof-writing skills, and skills that enable them to better read, understand, and communicate mathematics.

They will have learned to interpret and analyze fundamental principles and theory concerning basic algebraic structures, including groups, product structures, generators and relations, homomorphisms, and isomorphisms.

They will be able to construct accurate, meaningful examples of algebraic structures, and relate theory from this course to previous mathematical knowledge.

Students should understand how to analyze and algebraically describe the symmetry of plane figures and how the symmetries of different figures are related.

They will be able to prove and analyze the truth of algebraic statements, and to clearly communicate mathematical ideas, in both written and oral form.

AS A CAPSTONE: Students will develop and deepen their algebraic thinking in this 3rd course of the MAT 150 sequence. They will learn about fields as well as representation theory, which is an algebraic way of understanding symmetry, such as the symmetries of the physical universe. Via this subject area expertise, they will gain mastery in this area of specialization and improve their ability to communicate mathematics at a capstone level, commensurate with that expected of one with an undergraduate degree in mathematics.