Alphabeta MathDecomposition Inertia and Frobenius
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Absolute Values Completions and P Adic Numbers
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analyticity of Holomorphic Functions; Liouville and Morera
- Arc Length and Rectifiable Curves
- Artinian Rings and Length
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Variation and the Riemann–Stieltjes Integral
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Contour Integration
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Dedekind Domains and Ideal Classes
- 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
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Fields and Cyclotomic Extensions
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Integral Extensions and Going Up
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Number Fields Rings of Integers and Discriminants
- Order, Zorn's Lemma, and the Axiom of Choice
- Partitions of Unity and Paracompactness
- pi: the Equivalent Characterizations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Prime Ideal Decomposition Ramification and the Different
- Prime Spectra and Radicals
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
- Tensor Products of Modules
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Valuation Rings and Discrete Valuation Rings
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Theorem
Completion of an absolutely valued field
Definition
Normed vector space over an absolutely valued field
Theorem
Finite dimensional norm equivalence over a complete valued field
Lemma
Uniqueness of an extended complete field absolute value
Lemma
Hensel factor lifting over a complete valued field
Lemma
Irreducible polynomial coefficients in a complete valuation ring
Theorem
Unique extension of a nonarchimedean absolute value
Theorem
Number field places classification
Definition
Completion of a number field at a prime
Lemma
Number field completions as local polynomial factors
Theorem
Galois action on primes above a prime is transitive
Corollary
Galois prime decomposition efg
Definition
Decomposition group of a prime
Theorem
Decomposition group and completion
Definition
Inertia group of a prime
Lemma
Lifting residue frobenius by galois conjugates
Theorem
Decomposition inertia exact sequence
Corollary
Orders of decomposition and inertia groups
Theorem
Conjugacy of decomposition and inertia groups
Theorem
Decomposition and inertia in towers
Theorem
Decomposition and inertia fixed fields
Definition
Arithmetic frobenius coset
Theorem
Unramified frobenius element exists uniquely
Theorem
Frobenius elements above a prime are conjugate
Theorem
Frobenius order is residue degree
Corollary
Complete splitting and trivial frobenius
Corollary
Frobenius compatibility in finite towers
Lemma
Good polynomial reduction kills inertia
Theorem
Frobenius cycle type and prime splitting
Sources
- §5, Definitions 5.1–5.2 and Theorem 5.3, pp.8–9
- §5, Definition 5.4, p.9
- §5, Theorem 5.5 and proof, p.9
- §6, Lemma 6.1, pp.10–11
- §6, Theorem 6.5 and full proof, pp.12–13
- §6, proof of Theorem 6.4, p.11
- §6, Lemma 6.1 and Theorem 6.4, pp.10–11; Milne Theorem 7.38 for discrete separable specialization
- §7, Corollary 7.3 and preceding normalization, pp.15–16; Milne Theorem 7.14
- Chapter 7, Theorem 7.38 and Remark 7.39; Chapter 8, Proposition 8.2
- Chapter 8, Propositions 8.1–8.2, pp.135–136; Conrad Lemma 7.2, pp.14–15
- §9.2, Theorem 9.2.2 proof, pp.101–102
- §9.2, Theorem 9.2.2
- §9.3, Definition 9.3.1, p.104
- Chapter 8, Proposition 8.10, p.139
- §9.3.2, Definition 9.3.6 and Proposition 9.3.8, p.106
- Chapter 8, Frobenius element, footnote 1 on p.141; Stein Theorem 9.3.5
- §9.3.2, Theorem 9.3.5 (not design locator 9.3.2), p.106
- §9.3.2, Corollary 9.3.7, p.106
- §9.3, Lemma 9.3.2, p.104
- Chapter 8, Proposition 8.13, p.141
- Chapter 8, Proposition 8.11, p.140; Stein Propositions 9.3.3–9.3.4, pp.104–105
- Chapter 8, Frobenius element, p.141
- Chapter 8, Frobenius element, pp.141–142
- Chapter 8, Proposition 8.14; Stein Proposition 9.4.1
- Chapter 8, Frobenius element, p.142
- Chapter 8, p.142, paragraph after Proposition 8.14
- Chapter 8, Propositions 8.15–8.17, p.142
- Chapter 8, Proposition 8.21 and Theorem 8.23 proof, pp.144–145
- Chapter 8, Proposition 8.21, Corollary 8.22, Theorem 8.23, pp.144–145