Almost Everywhere
← Back

collection / modern mathematics

Week 01 — Field Extensions and Galois Theory

algebragalois theory

Week 01 — Field Extensions and Galois Theory

Central Question

Why did Galois theory have to exist?

The Story

Follow the need to construct new number systems into the symmetries of polynomial roots and the obstruction to radical formulas.

Polynomial→Field Extension→Splitting Field→Automorphisms→Galois Group\text{Polynomial}\rightarrow\text{Field Extension}\rightarrow\text{Splitting Field}\rightarrow\text{Automorphisms}\rightarrow\text{Galois Group}

How to Use These Notes

Each session now contains a substantive Mathematical Notes section: definitions, derivations, worked examples and necessary caveats. The original dialogue prompt remains available, and the personal Ideas, Questions and reflections sections are yours to edit.

The scope is a complete first pass through classical finite Galois theory in characteristic zero, with the positive-characteristic separability caveat included. It is not a condensation of every chapter of Stewart's book.

Recurring Examples

  • Q(2)\mathbb Q(\sqrt2): constructing a field and its first symmetry.
  • Q(2,3)\mathbb Q(\sqrt2,\sqrt3): tower degrees, independent signs and a complete fixed-field table.
  • x3−2x^3-2: a nonnormal root field, a degree-six splitting field and a nonabelian Galois group.
  • x5−6x+3x^5-6x+3: a concrete obstruction to solving by radicals.

Reading Companion

Ian Stewart, Galois Theory, fifth edition. Read Chapters 4–6 for extensions and degree, Chapters 8–13 for symmetries and the correspondence, Chapters 14–15 for solvability, and §18.4 for the converse radical criterion. Each session supplies more precise references. Publisher information.

Sessions

  1. Adjoining a Root
  2. Why Quotient by a Polynomial?
  3. When Different Constructions Give the Same Field
  4. Algebraic Numbers and Minimal Polynomials
  5. Tower Extensions
  6. Splitting Fields
  7. Field Automorphisms
  8. From Automorphisms to Groups
  9. My First Genuine Galois Group
  10. Fundamental Theorem of Galois Theory
  11. Why the Quintic Changed Mathematics
  12. Week 1 Synthesis

Connections Beyond This Week

Month overview · Exploration Map · Home