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.
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Suggested Schedule:
Lecture(s) 
Sections 
Comments/Topics 
Week 1 
Definition of Bilinear Forms (Note: Math 67 covers inner products); Symmetric Forms – orthogonality the Geometry associated to a symmetric form. 

Week 2 
Hermitian Forms (Note: Math 67 covers Hermitian matrices); The Spectral Theorem. 

Week 3 
The classical linear groups; The special unitary group (Note: Math 67 covers unitary matrices); Orthogonal representation of SU2. 

Week 4 
SL(2); Abstract Fields; Matrix groups and linear algebra over abstract fields (from Chap. 3 of Artin); Definition of Rings. 

Week 5 
Formal construction of integers and polynomials; Homomorphisms and Ideals; Quotient rings and relations in a ring. 

Week 6 
Integral domains and fraction fields; Maximal ideals; Factorization of integers and polynomials. 

Week 7 
Unique factorization domains, principal ideal domains, and Euclidean domains; Gaussian integers; Primes. 

Week 8 
Ideal factorization; Definition of modules; Matrices, free modules and bases. 

Week 9 
Diagonalization of integer matrices; Generators and relations for modules; Structure theorem for Abelian groups. 

Week 10 
Application to linear operators. 
Additional Notes:
If there is extra time, continue with Chapter 11 of Artin (modules), or go back and fill in topics such as: 8.6 The Lie Algebra; 8.7 Simple Groups; 10.5 Adjunction of Elements.
Learning Goals:
Throughout the course, students will write proofs of mathematical statements of increasing complexity. They will develop the ability to reach conclusions by reasoning logically from first principles and to justify those conclusions in clear, persuasive language (either oral or written).
They will gain proficiency in abstraction, rigorously developing algebra from existing theory (and from axioms, definitions, undefined terms, etc.) in accordance with axiomatic method.
Students will learn how algebraic manipulations they have made in earlier courses follow the same familiar pattern in this course. This is a very important unifying objective of abstract algebra.
A major goal is that students thoroughly learn the definitions of the concepts studied and will have an intuitive idea of what any definition means; knowing some examples and nonexamples, and having some practical skill in working with the definition in mathematical arguments.
Students will ask (and answer) their own investigative questions such as:
 Can I find a good example or counterexample of this?
 Is that hypothesis in that theorem really necessary? What happens if I drop it?
 Does this property imply that property, and, if not, can I find a counterexample?