Mathematics / advanced
Field extensions and Galois symmetry
Relate polynomial equations to field extensions, automorphism groups, solvability, and finite-field structure.
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Inside this module
Module 01 advanced
Field extensions and Galois symmetry
Relate polynomial equations to field extensions, automorphism groups, solvability, and finite-field structure.
- 01
Field extensions: adjoining a number without losing arithmetic
Distinguish algebraic and transcendental elements, construct a generated field, and calculate inside a quadratic extension.
24 min - 02
Minimal polynomials: the first relation determines the dimension
Prove irreducibility of a minimal relation, identify a power basis, and reduce arithmetic in a simple extension.
26 min - 03
Towers and linear disjointness: when adjoining information multiplies
Prove the tower law with product bases and diagnose when two field extensions contribute independent coordinates.
29 min - 04
Splitting fields: put every conjugate in the same arithmetic universe
Construct a minimal field containing all roots, distinguish it from a root field, and understand algebraic closure.
26 min - 05
Separability: count distinct conjugates, not repeated copies
Detect repeated roots with a formal derivative and explain how positive characteristic permits algebraic degree without distinct conjugates.
28 min - 06
Normal Extensions and Field Automorphisms
Understand normality as closure under conjugates, and distinguish embeddings from automorphisms.
28 min - 07
The Galois Correspondence
Translate intermediate fields into stabilizing subgroups, with degrees, reversed inclusion, and normal quotients.
30 min - 08
Finite Fields and Frobenius
Build finite fields from irreducible polynomials and use the Frobenius map to read their Galois groups and subfields.
29 min - 09
Cyclotomic Extensions and Constructibility
Connect roots of unity, modular unit groups, and the quadratic towers behind straightedge-and-compass constructions.
31 min - 10
Solvable Groups and Solvability by Radicals
Translate radical formulas into solvable Galois groups, and distinguish the general quintic obstruction from special solvable examples.
33 min