Introduction to Galois Theory
Historical Context
Galois theory explains why there is no general formula for solving polynomial equations of degree (Abel-Ruffini theorem).
Field Extensions
Separable Extensions
Galois Extension
Galois Group
Fundamental Theorem of Galois Theory
Example:
where
This group is , matching the intermediate fields: .
Solvability by Radicals
Abel-Ruffini Theorem
Splitting Field
Examples
Cubic: over has Galois group (solvable, so solvable by radicals).
Quintic: over has Galois group (not solvable).
Cyclotomic Extensions
where .