Spring Quarter 2018, Math 220C, Modern Algebra III


Instructor:
A. Agboola
Lecture:
Tu--Th 9:30am-10:45am, SH 1609

Office:
6724 South Hall
Office hours:
Tues.--Thurs. 11:00am--12:30pm
Textbooks:
D. Dummit, R. Foote, Abstract Algebra (Third Edition), Wiley, (2004) (required).
S. Lang, Algebra (Revised Third Edition), Springer, (2002) (required).


Homework:
Homework may be assigned, and some of it may be collected. Your work should be typed neatly in TeX on letter-sized paper.
The following link will take you to the homework problems:

  • Problem sheets


    • Examinations:
    There will be one take-home midterm examination. It will be handed out in class on Thursday, May 3, and will be due via Gauchospace by 12 noon on Monday, May 7. Details regarding the final examination for the course will be given later.


    PLEASE NOTE THAT NO MAKEUP EXAMINATIONS WILL BE GIVEN IN THIS COURSE.
    Course Outline:
    We shall aim to cover the following topics. Additional topics will be covered if time permits.

    Algebraic and transcendental field extensions. Tower laws. Existence and uniqueness of the splitting field of a polynomial. The primitive element theorem. Existence and uniqueness of the algebraic closure of a field. Ruler-and-compass constructions.

    Normal extensions. Automorphism groups of fields; fixed fields. The fundamental theorem of Galois theory.

    Finite fields. Cyclotomic fields.

    Cyclic extensions and extraction of roots. Soluble groups and equations soluble by radicals. Solution of equations of degree 3 and $4$; insolubility of the general quintic equation.

    Linear representations of groups, matrix representations. Equivalence of representations, invariant subspaces and irreducibility, complete reducibilty of representations. Uniqueness of decompositions into irreducible components. Schur's lemma. Characters, orthogonality relations. Determination of a completely reducible representation, up to equivalence, by its character. Conjugacy classes, number of irreducible representations. The group algebra, tensor product of representations. Tensor, polynomial, and exterior algebras. Induced representations and the Frobenius reciprocity theorem.