Glossary

12.3 Decomposition and inertia groups

3 min read•july 30, 2024

Decomposition and inertia groups are key players in understanding prime behavior in Galois extensions. They measure splitting and , giving us insight into how primes behave when we extend our number fields.

These groups are essential tools in ramification theory, connecting local and global perspectives. By studying their structure and relationships, we can unravel the mysteries of factorization and residue field extensions in algebraic number theory.

Decomposition and Inertia Groups

Definitions and Basic Properties

Top images from around the web for Definitions and Basic Properties

  • Théorie de Galois à l'origine — Wikipédia View original

    Is this image relevant?

  • algebraic number theory - On Hilbert Class Polynomial - Mathematics Stack Exchange View original

    Is this image relevant?

  • Théorie de Galois à l'origine — Wikipédia View original

    Is this image relevant?

1 of 3

Top images from around the web for Definitions and Basic Properties

  • Théorie de Galois à l'origine — Wikipédia View original

    Is this image relevant?

  • algebraic number theory - On Hilbert Class Polynomial - Mathematics Stack Exchange View original

    Is this image relevant?

  • Théorie de Galois à l'origine — Wikipédia View original

    Is this image relevant?

1 of 3
  • D(P|p) of a prime ideal P in Galois extension L/K fixes P and measures splitting behavior
  • I(P|p) fixes the residue field extension and measures ramification
  • D(P|p) acts transitively on prime ideals of L lying over p
  • Fixed field of D(P|p) corresponds to decomposition field of extension for prime p
  • Fixed field of I(P|p) corresponds to inertia field of extension for prime p
  • Decomposition group contains both ramification and splitting information
  • Inertia group focuses solely on ramification

Examples and Applications

  • In quadratic extension Q(2)/Q\mathbb{Q}(\sqrt{2})/\mathbb{Q}, prime 2 ramifies with D(P|2) = I(P|2) = Gal(Q(2)/Q\mathbb{Q}(\sqrt{2})/\mathbb{Q})
  • For extension Q(ζ7)/Q\mathbb{Q}(\zeta_7)/\mathbb{Q} with ζ7\zeta_7 a 7th root of unity, prime 7 is totally ramified
  • In cyclotomic extension Q(ζp)/Q\mathbb{Q}(\zeta_p)/\mathbb{Q} for prime p, D(P|p) is cyclic of order p-1
  • extension Qp(pn)/Qp\mathbb{Q}_p(\sqrt[n]{p})/\mathbb{Q}_p has D(P|p) = Gal(Qp(pn)/Qp\mathbb{Q}_p(\sqrt[n]{p})/\mathbb{Q}_p)

Relationship of Decomposition and Inertia Groups

Group Structure and Quotients

  • I(P|p) of D(P|p)
  • Quotient D(P|p)/I(P|p) isomorphic to Galois group of residue field extension
  • Index [D(P|p) : I(P|p)] equals degree of residue field extension
  • Order |D(P|p)| product of ramification index, residue degree, and degree of splitting field
  • Order |I(P|p)| equals ramification index of p in L/K
  • Relationship expressed as |D(P|p)| = e(P|p) · f(P|p) · g(P|p) (e ramification index, f residue degree, g number of prime ideals over p)

Examples of Group Relationships

  • Unramified prime in Q(5)/Q\mathbb{Q}(\sqrt{5})/\mathbb{Q}: I(P|p) trivial, D(P|p) cyclic of order 2
  • Totally ramified prime in Q(23)/Q\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}: D(P|2) = I(P|2) of order 3
  • Split prime in Q(−1)/Q\mathbb{Q}(\sqrt{-1})/\mathbb{Q}: D(P|p) trivial, I(P|p) trivial
  • Tamely ramified prime in Q(ζp)/Q\mathbb{Q}(\zeta_p)/\mathbb{Q}: I(P|p) cyclic of order p-1

Structure of Decomposition and Inertia Groups

Specific Extension Types

  • Unramified primes inertia group trivial, decomposition group cyclic (Frobenius automorphism generator)
  • Totally ramified extensions decomposition and inertia groups coincide, order equals ramification index
  • Tamely ramified extensions inertia group cyclic, order equals ramification index
  • Wildly ramified extensions inertia group complex structure, often p-groups (p characteristic of residue field)
  • Cyclotomic extensions analyzable using cyclotomic field theory and ramification behavior
  • Local field extensions decomposition group entire Galois group, inertia group structure related to local class field theory

Examples in Various Extensions

  • Q(−5)/Q\mathbb{Q}(\sqrt{-5})/\mathbb{Q}: prime 2 inert, D(P|2) cyclic of order 2, I(P|2) trivial
  • Q(ζ8)/Q\mathbb{Q}(\zeta_8)/\mathbb{Q}: prime 2 totally ramified, D(P|2) = I(P|2) of order 4
  • Qp(ζp)/Qp\mathbb{Q}_p(\zeta_p)/\mathbb{Q}_p: D(P|p) = Gal(Qp(ζp)/Qp\mathbb{Q}_p(\zeta_p)/\mathbb{Q}_p), I(P|p) of order p-1
  • Fp((t))(t1/p)/Fp((t))\mathbb{F}_p((t))(t^{1/p})/\mathbb{F}_p((t)): wildly ramified, I(P|p) elementary abelian p-group

Ramification in Galois Extensions

Analysis Tools and Applications

  • Decomposition and inertia groups analyze ramification of prime ideals in Galois extensions
  • Ramification index of prime p in L/K equals order of inertia group I(P|p)
  • Prime p splitting in L/K determined by decomposition group D(P|p) structure
  • Higher ramification groups (subgroups of inertia group) provide wildly ramified extension depth information
  • Hilbert different and discriminant computable using inertia groups of ramified primes
  • Groups crucial in Hilbert-Speiser theorem formulation and proof for tamely ramified abelian extensions
  • Essential for understanding prime behavior in tower extensions and ramification-Galois theory relationship

Concrete Examples and Computations

  • Q(23)/Q\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}: prime 2 ramification index 3, |I(P|2)| = 3
  • Q(ζ7)/Q\mathbb{Q}(\zeta_7)/\mathbb{Q}: prime 7 totally ramified, ramification index 6, |I(P|7)| = 6
  • Q2(2)/Q2\mathbb{Q}_2(\sqrt{2})/\mathbb{Q}_2: wildly ramified, higher ramification groups G₀ = G₁ = I(P|2), G₂ = {1}
  • Cyclotomic extension Q(ζpn)/Q\mathbb{Q}(\zeta_{p^n})/\mathbb{Q}: prime p ramification index pn−1(p−1)p^{n-1}(p-1), |I(P|p)| = pn−1(p−1)p^{n-1}(p-1)

Key Terms to Review (13)

Chebotarev Density Theorem: The Chebotarev Density Theorem describes the distribution of prime ideals in a number field and how they split in finite Galois extensions. It connects the splitting behavior of primes to the structure of Galois groups, providing a way to determine the density of primes that behave in certain ways relative to these extensions.
Decomposition Group: A decomposition group is a subgroup of the Galois group that describes how a prime ideal in a number field splits into prime ideals in an extension field. It provides insight into the behavior of prime ideals under field extensions, revealing whether a prime remains irreducible or decomposes into several factors. Understanding decomposition groups is crucial for analyzing ramification and inertia, especially when exploring the structure of field extensions.
Degree of extension: The degree of extension refers to the dimension of a field extension as a vector space over its base field. It measures how many elements from the larger field can be expressed as linear combinations of elements from the smaller field. Understanding the degree of extension is crucial for exploring properties like algebraic closures, norms, traces, and various structures in number theory.
Emil Artin: Emil Artin was a prominent 20th-century mathematician known for his significant contributions to algebraic number theory, particularly in the areas of class field theory and algebraic integers. His work has influenced various aspects of modern mathematics, linking concepts like field extensions and ideals to the broader framework of number theory.
Finite Extension: A finite extension is a field extension in which the larger field has a finite dimension as a vector space over the smaller field. This concept connects various aspects of algebraic structures, showcasing how algebraic numbers and integers can form fields with finite degrees, and how properties such as norms, traces, and discriminants are integral to understanding these extensions.
Inertia group: An inertia group is a subgroup of the Galois group that describes how primes split in a field extension and identifies the behavior of the ramification of those primes. It plays a crucial role in understanding how primes behave under various extensions, particularly in relation to decomposition and ramification. Inertia groups provide insight into the local behavior of field extensions, connecting to concepts like Frobenius automorphisms and completions of number fields.
Isomorphism: Isomorphism refers to a structural similarity between two algebraic structures that allows for a one-to-one correspondence between their elements while preserving the operations defined on those structures. This concept is essential in various areas of mathematics, as it highlights the inherent equivalence between different algebraic systems, showing how they can behave identically despite potentially differing appearances.
Kummer's Theorem: Kummer's Theorem provides a way to understand the behavior of primes in the context of cyclotomic fields, specifically relating to how these primes split in these fields. It highlights the relationship between the ramification of primes and their decomposition into distinct prime factors, which is crucial when studying the structure of cyclotomic fields and understanding the decomposition and inertia groups associated with primes in number theory.
Local Field: A local field is a complete discretely valued field that is either finite or has a finite residue field. Local fields play a crucial role in number theory as they provide a framework to study properties of numbers in localized settings, allowing for techniques such as completion and the analysis of primes in extensions.
Normal Subgroup: A normal subgroup is a subgroup that is invariant under conjugation by members of the group, meaning that for every element in the group, when you conjugate any element of the subgroup by that group element, the result stays within the subgroup. This property is crucial because it allows for the construction of quotient groups, which are fundamental in understanding group structure and symmetry. Additionally, normal subgroups play a key role in the context of Galois theory, where they relate to the decomposition and inertia groups associated with field extensions.
Prime Ideal: A prime ideal in a ring is a special type of ideal such that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal. This concept helps to identify the structure and properties of rings, especially in understanding the relationships between ideals and factors in rings, connecting closely with notions like unique factorization.
Ramification: Ramification refers to how primes in a base field split or remain inert when extended to a larger field. It highlights the behavior of prime ideals under field extensions, particularly focusing on their splitting, degree of extension, and how they relate to the discriminant. This concept is crucial for understanding the structure of number fields and how they behave under various algebraic operations.
Richard Brauer: Richard Brauer was a prominent mathematician known for his significant contributions to group theory, representation theory, and algebraic number theory. His work laid the foundation for understanding the relationships between various algebraic structures, particularly through the study of decomposition and inertia groups in Galois theory, which are vital for analyzing how primes behave in extensions of number fields.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide