Alphabeta MathGalois Orbits and Descent of Simple Finite-Group Modules: Examples
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1 · Prerequisites
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Galois Orbits and Descent of Simple Finite-Group Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Semidirect Products, Automorphism Groups and Split Extensions
- Simple Field Extensions and the Construction of the Complex Numbers
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
- The ZFC Axioms and the Basic Set Constructions
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Example
Descent of the two nontrivial characters of C₃
Example
The rational simple quaternion block
Counterexample