Second Course in Abstract Algebra

Instructor: Vera Serganova
Email address: serganov@math

webpage:/~serganov

Phone Number: 642-2150

Office hours: Tu,Th 5:30-6:30 in 709 Evans

Text: E. Artin, Galois theory

Homework: Each Thursday problems will be posted on this webpage. Homework will be due next Thursday

Exams: There will be one midterm on Thursday, March 2 during usual class hours. Usually we will have one quiz every second week

Final Exam is on Friday, May 20.

Grading policy: Your grade will be computed according to the following rule: 20% for your homework, 20% for quizzes, 20% for the midterm and 40% for the final.

Attention: change of date for the midterm. It will be on March 7 instead of March 2

Course outline

Review of groups

Definitions. Examples. Lagrange's Theorem. Conjugation. Normal Subgroups

Abelian groups

Permutation groups, Cayley theorem

Action, orbits, counting formulas

Solvable and simple groups

Linear algebra

Vector spaces, basis, dimension

Trace and determinant

Fields and Galois theory

Algebraic extensions. Degree.

Polynomials. Uniqueness of decomposition into the product of irreducible polynomials

Splitting field

Normal and separable extensions

Galois group

Fundamental theorem of Galois theory

Finite fields

Roots of unity (cyclotomic extensions)

Kummer fields

Simple extension. Existence of a primitive element in separable extensions

Normal basis

Algebraic closure

Applications

Solution of algebraic equations in radicals

Equations of the third and forth degree

Galois group of a general equation

Equations of prime degree

Ruler and compass construction

Attension: office hours week of May 13: Tu,Th 4-6:30

Notes on polynomials of degree 3 and 4