Ben-Gurion University of the Negev
Department of Mathematics
FIELD AND GALOIS THEORY
Course number: 201-17041; Spring Semester 2016
Instructor:
Ido Efrat
- Office: Mathematics Building, Room 106
- Office hours: Monday 10-12
- Tel.: (08) 6461627
- e-mail: efrat@math.bgu.ac.il
Time and place:
Monday 12-14, Building 90, Room 242
Wednesday 13-15, Building 58, Room 101
Prerequisite: Algebraic Structures
Exercises:
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Some practice problems
Course Plan:
- Fields
- Basic facts and examples
- Prime fields and the characteristic
- Decomposition of polynomials
- Gauss' lemma
- Eisenstein polynomials
- Field extensions
- The tower property
- Adjoining an element to a field
- Algebraic and transcendental extensions
- Transcendental numbers
- Constructions using a ruler and a compass
- Constructible numbers
- Basic constructions
- The unsolved problems of Greek geometry:
- why cann't we square the circle?
- the duplication of the cube
- trisection of angles
- regular heptagons
- Gauss' construction of a regular 17-edge polygon
- Fermat primes
- Algebraicaly closed fields
- Equivalent conditions
- Examples
- Existence and uniqueness
- Splitting fields
- Splitting of polynomials
- Existence and size
- Extensions of homomorphisms
- Galois extensions
- Automorphisms
- Normality
- Separability
- Fixed fields
- Artin's theorem
- Galois groups
- The fundamental theorem of Galois theory
- Cyclic extensions
- Dedekind's theorem on independence of characters
- The structure of cyclic extensions
- Solvability of polynomials by radicals
- Equations of degrees 2,3,4: the Tartaglia-Cardan and Ferrari formulas
- The Galois group of a polynomial
- Solvable groups
- Galois' theorem
- The insolvability of polynomials of degree >4
- Finite fields
- Existence and uniqueness
- Galois groups over finite fields
- The multiplicative group
- Roots of unity
- Cyclotomic extensions
- The cyclotomic polynomials
- Geometric constructions again
- Gaussian periods (if time allows)
- The fundamental theorem of algebra (if time allows)
- Ordered and real closed fields
- A Galois-theoretic proof of the theorem
- The Artin-Schreier Theory
Exams and Grading:
There will be a final exam.
The course' grade will be the final exam's grade.
Recommanded books
- D. Cox, Galois Theory, Wiley Interscience, 2004
- J.-P. Escofier, Galois Theory, Springer, 2001
- D.J.H. Garling, A Course in Galois Theory, Cambridge University Press
- N. Jacobson, Basic Algebra, Freeman 1974
- S. Lang, Algebra, Addison-Weseley, 1965
- M. Artin, Algebra, Prentice-Hall, 1991
- J.S. Milne, Fields and Galois Theory, On-line lecture notes, with exercises.
- J. Swallow, Exploratory Galois Theory, Cambridge University Press 2004
Some Links:
The life
and times of Evariste Galois
The 17-gon
Biographies of Cardano
and Tartaglia
Tartaglia vs. Cardano