 ## FIELD AND GALOIS THEORY

#### Course number: 201-17041; Spring Semester 2016 ### COURSE' ANNOUNCEMENT (in Hebrew)

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

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7

#### Course Plan:

1. Fields
• Basic facts and examples
• Prime fields and the characteristic

2. Decomposition of polynomials
• Gauss' lemma
• Eisenstein polynomials

3. Field extensions
• The tower property
• Adjoining an element to a field
• Algebraic and transcendental extensions
• Transcendental numbers

4. 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

5. Algebraicaly closed fields
• Equivalent conditions
• Examples
• Existence and uniqueness

6. Splitting fields
• Splitting of polynomials
• Existence and size
• Extensions of homomorphisms

7. Galois extensions
• Automorphisms
• Normality
• Separability
• Fixed fields
• Artin's theorem
• Galois groups
• The fundamental theorem of Galois theory

8. Cyclic extensions
• Dedekind's theorem on independence of characters
• The structure of cyclic extensions

9. 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

10. Finite fields
• Existence and uniqueness
• Galois groups over finite fields
• The multiplicative group

11. Roots of unity
• Cyclotomic extensions
• The cyclotomic polynomials
• Geometric constructions again
• Gaussian periods (if time allows)

12. The fundamental theorem of algebra (if time allows)
• Ordered and real closed fields
• A Galois-theoretic proof of the theorem
• The Artin-Schreier Theory

There will be a final exam. The course' grade will be the final exam's grade.

#### Recommanded books

1. D. Cox, Galois Theory, Wiley Interscience, 2004
2. J.-P. Escofier, Galois Theory, Springer, 2001
3. D.J.H. Garling, A Course in Galois Theory, Cambridge University Press
4. N. Jacobson, Basic Algebra, Freeman 1974
5. S. Lang, Algebra, Addison-Weseley, 1965
6. M. Artin, Algebra, Prentice-Hall, 1991
7. J.S. Milne, Fields and Galois Theory, On-line lecture notes, with exercises.
8. J. Swallow, Exploratory Galois Theory, Cambridge University Press 2004