### 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