Author(s): Richard Koch
Year: 2017
Commentary: Downloaded from https://pages.uoregon.edu/koch/Galois.pdf
Preliminaries
The Extension Problem; Simple Groups
An Isomorphism Lemma
Jordan Holder
The Symmetric and Alternating Groups
The Quadratic, Cubic, and Quartic Formulas
The Quadratic Formula
The Cubic Formula
The Quartic Formula
Field Extensions and Root Fields
Motivation for Field Theory
Fields
An Important Example
Extension Fields
Algebraic Extensions; Root Fields
Irreducible Polynomials over Q
The Degree of a Field Extension
Existence of Root Fields
Isomorphism and Uniqueness
Putting It All Together
Splitting Fields
Factoring P
The Splitting Field
Proof that Splitting Fields Are Unique
Uniqueness of Splitting Fields, and P(X) = X3 - 2
Finite Fields
Finite Fields
Beginning Galois Theory
Motivation for Galois Theory
Motivation; Putting the Ideas Together
The Galois Group
Galois Extensions and the Fundamental Theorem
Galois Extensions
Fundamental Theorem of Galois Theory
Important Note
An Example
Finite Fields Again
Concrete Cases of the Theory
Cyclotomic Fields
Galois Group of a Radical Extension
Structure of Extensions with Cyclic Galois Group
Solving Polynomials by Radicals
Solving Polynomial Equations with Radicals
The Flaw
The Fix
Galois' Theorem on Solving Via Radicals
Solving Generic Equations by Radicals
A Polynomial Equation Which Cannot Be Solved by Radicals
Straightedge and Compass Constructions
Constructions With Straightedge and Compass
Complex Extensions and Constructions
Trisecting Angles; Doubling the Cube
Irrationality and Transcendence of and e
Irrationality e and
Transcendence of e
Intermission: Fundamental Theorem of Symmetric Polynomials
Transcendence of
Special Cases
The Discriminant
The Cubic Case
Cyclotomic Fields
Galois Group of Xp - a over Q
Constructing Regular Polygons
Constructing Regular Polygons
Normal and Separable Extensions
Normal Extensions, Separable Extensions, and All That
Galois Theory and Reduction Modulo p
Preview
X5 - X - 1
A Tricky Point Concerning Straightedge and Compass Constructions
Polynomials with Galois Group Sn
Every Finite G is a Galois Group
Proof of Dedekind's Theorem
