Galois Theory 2019

Tutorial I  (by Asuka Kumon)

• Reviews on rings and ideals

Tutorial II (by Asuka Kumon)

• Irreducible criterion for polynomials over Q

Tutorial III   

• Field extensions and their degrees :  Tower law,

• Strategy in field theory, 

• Algebraic extensions and elements

Tutorial IV 

• Splitting fields, 

• Field embeddings,

• Concrete examples : biquadratic extensions and cyclotomic (p-th and 8-th) extensions

Tutorial V

• Separable polynomials and separable extensions 

• Finite fields : exponential dream and separability

• Inseparable extensions : examples from function field

Tutorial VI

• Galois extensions : normal & separable

• Examples : biquadratic extensions and Kummer extensions

• Degree 2 extensions are Galois in char=/=2

Tutorial VII

• Fundamental theorem of Galois theory

• Examples : 8th cyclotomic, biquadratic and dihedral extensions

• Computations of Galois groups

Tutorial VIII

• Galois groups of polynomials and transitivity

• Galois groups of cubics

• Galois groups of quartic

Tutorial IX

• Finite fields as splitting fields of x^{p^n}-x

• Finite fields via irreducible polynomials

• Galois theory of finite fields : cyclicity of Galois group and Frobenius automorphism

Tutorial X

• Solvable extensions by radicals

• Solvable groups and extensions

• An example of insolvable quintic