Graduate Algebra

Instructor: Guanghua Ji

Textbook: R. B. Ash, Basic abstract algebra: for graduate students and advanced undergraduates, Dover Publications, 2006.

Classroom and Time: C701, 5-7, Friday

Description and Prerequisites: This is a standard graduate course in abstract algebra. We study fundamental algebraic structures, namely groups, rings, modules and fields, and maps between these structures. In addition, we will also cover some basic group representation theory. The only prerequisites listed for this course are primary arithmetic and linear algebra.

Grading: The course grade will be based on attendance(10%), homework(20%) and final exam(70%).

Office Hours: Open, Come any time! B1042

References:

  1, M. Artin, Algebra, Addison Wesley, 2010.

  2, D. S. Dummit and R. M. Foote, Abstract algebra(3rd),             John Wiley and Sons, Inc, 2004.

  3, W. Fulton and J. Harris, Representation Theory, A First      Course, Springer, GTM 129, 1991.

  4, J. P. Serre, Linear representations of finite groups,      GTM 42, Springer.

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Syllabus:

1, Groups and groups actions

2/25, Groups, subgroups, cyclic groups, order, permutation groups. Homework: 1.1: 3,6,8,9,14; 1.2: 4,5,6,8.

3/4, cosets, Lagrange's theorem, normal subgroup, quotient group. Homework: 1.3: 3,6,7,8,13.

3/11, Isomorphism theorem, first, second, third, direct products. Homework: 1.4: 3,5,8; 1.5: 1,3,6.

3/18, Cayley's theorem, groups acting on sets, conjugation of subgroups, normalizer of subgroups, the orbit-stabilizer theorem. Homework: 5.1: 2,4, 5,8; 5.2: 2,3,6.

3/25, Sylow theorems, Cauchy's theorem, simple groups, applications. Homework: 5.4: 1,4,8,9; 5.5: 1,3.

4/1, Composition series, Jordan-Holder theorem, solvable groups, nilpotent groups, free groups. Homework: 5.6: 4,5,6,7,8,9; 5.7: 1,4,7; 5.8: 2.

2, Basic representation theory

4/8, Representations of finite groups: definitions, subrepr, irreducible repr, completely reducible, unitary repr, Maschke's theorem. Homework.

4/15, Schur's lemma: group algebra, Schur orthogonality relations. Homework.

4/22, Characters theory: class functions, first(second)orthogonality relatioins, regular representations, character tables. Homework.

3, Ring, Field and Module theory

4/29, Basic definitions: Ring, integral domain, division ring, field, characteristic, subring, ideals, contruction of quotient rings. Homework: 2.1: 2,4,5; 2.2: 6,8.

5/6, Maximal ideals, prime ideals, polynomial rings. Homework: 2.4: 2,5,7.

5/13, UFD, PID, Euclidean domains, fraction fields. 2.6: 7,8; 2.7: 2,8.

5/20, Field extensions, splitting fields, algebraic closures. Homework: 3.1: 3,4; 3.2: 2,3.

5/27, Algebraic closures, normal extensions, module.

4, Galois theory

6/3, Fixed fields, Galois groups, the fundamental theorem, computing a Galois groups.

6/10, Finite fields, cyclotomic fields.

6/17, Cyclic and Kummer extensions, solvability by radicals.

Graduate Algebra Final Exam

other online resources

All are welcome to attend!