12.2 Ramification groups and higher ramification theory
4 min read•july 30, 2024
Ramification groups and theory dive deep into the structure of extensions. They provide a powerful toolset for understanding how prime ideals behave when moving from a base field to an extension field.
This topic builds on the foundation of valuations, expanding our understanding of field extensions. By examining ramification indices and inertia degrees, we gain crucial insights into the intricate relationships between local fields and their extensions.
Ramification in Field Extensions
Fundamentals of Local Field Extensions
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Ramification in local field extensions describes prime ideal behavior when transitioning from a base field to an extension field
Local fields are complete fields with respect to a
Typically finite extensions of p-adic numbers Qp
Fields of formal Laurent series
e measures the degree of prime ideal splitting or ramification in an extension
Extension L/K of local fields categorized as:
Unramified when e = 1
Ramified when e > 1
f represents the degree of the residue field extension
Fundamental equality established: [L:K]=efg
g denotes the number of prime ideals in the extension lying over the prime ideal in the base field
Key Concepts and Measurements
Prime ideal behavior crucial in understanding structure of field extensions
Discrete valuation provides a way to measure "size" of elements in the field
Ramification index e calculated as:
e=vL(πK)
πK is a uniformizer of the base field K
vL is the valuation in the extension field L
Inertia degree f computed as:
f=[kL:kK]
kL and kK are residue fields of L and K respectively
Examples of ramification:
Quadratic extension Q2(√2)/Q2 (ramified)
Qp(ζp-1)/Qp, where ζp-1 is a primitive (p-1)th root of unity
Ramification Groups and Properties
Definition and Structure
Ramification groups form a decreasing filtration of the for finite Galois extensions of local fields
Lower ramification groups Gi defined for i ≥ -1
G-1 represents the full Galois group
G0 is the inertia group
Upper ramification groups Gv defined using Hasse-Herbrand function φ
Relation: Gv=Gϕ(v)
Fixed fields of ramification groups create a tower of extensions
Each level provides more detailed ramification information
Quotients Gi/Gi+1 are elementary abelian p-groups
p is the residue characteristic
Structure of G0/G1 isomorphic to a subgroup of the multiplicative group of the residue field
Properties and Applications
Ramification groups measure "wildness" of ramification
Lower numbering (Gi) convenient for subfields
Upper numbering (Gv) well-behaved for quotients
Herbrand's theorem relates lower and upper numbering systems
Applications in local class field theory and Galois representations
Examples:
In Qp(ζp)/Qp: G0 = G1 has order p-1, and G2 = {1}
For a totally of degree p: G0 = G1 has order p, and G2 = {1}
Higher Ramification Groups
Advanced Concepts and Functions
Higher ramification groups enable refined analysis of wildly ramified extensions
Herbrand function ψ introduced as inverse of Hasse-Herbrand function φ
Relationship between lower and upper numbering established:
Gv=Gψ(v) and Gi=Gϕ(i)
Behavior of ramification groups studied under:
Subextensions
Quotients
Herbrand theorem formalizes these relationships
Breaks in ramification filtration correspond to indices where ramification groups change
Hasse-Arf theorem states breaks in upper numbering are integers for abelian extensions
Analytical Tools and Examples
Herbrand function ψ defined as:
ψ(u)=∫0u(G0:Gt)dt
Hasse-Herbrand function φ is the inverse of ψ
Ramification breaks provide insight into extension structure
Examples of ramification breaks:
In Qp(ζp)/Qp: single break at v = 1
For wildly ramified quadratic extension of Q2: break at v = 2
Applications in studying:
Galois representations
L-functions
Iwasawa theory
Ramification Theory Applications
Classification and Analysis of Extensions
Ramification theory classifies extensions of local fields based on ramification behavior
Discriminant and different of an extension related to ramification groups
Provide measure of ramification
Local class field theory connected to ramification theory
Norm groups correspond to ramification filtration
Structure of totally ramified extensions analyzed using ramification groups
Particularly useful for cyclic extensions of degree p
Ramification theory applied to study:
Galois representations
Conductors (important in number theory and arithmetic geometry)
Local Kronecker-Weber theorem proved using ramification theory
Characterizes abelian extensions of Qp
Practical Implications and Examples
Ramification theory crucial in understanding:
Local zeta functions
Local Langlands correspondence
Applications in cryptography and coding theory
Design of error-correcting codes over p-adic fields
Examples of applications:
Computation of local epsilon factors in the theory of L-functions
Classification of extensions of Qp with given Galois group
Ramification theory in global settings:
Study of ramification in number fields
Analysis of reduction of elliptic curves
Key Terms to Review (17)
Artin's Theorem: Artin's Theorem states that for a given finite Galois extension of local fields, the higher ramification groups can be understood in terms of the Galois group of the extension. This theorem provides a critical link between the structure of the Galois group and the behavior of prime ideals in the extension, helping to classify the ramification of primes and their behavior under extension.
Complete discrete valuation field: A complete discrete valuation field is a field equipped with a discrete valuation that allows for the measure of the size of its elements and is complete in the sense that every Cauchy sequence converges within the field. This structure is fundamental in understanding the behavior of local fields and their extensions, particularly in relation to ramification, which deals with how primes split in extensions of fields.
Discrete valuation: A discrete valuation is a function defined on the field of fractions of a discrete valuation ring that assigns to each non-zero element a non-negative integer, representing the 'order' of that element. This concept helps in understanding the structure of local fields and plays a crucial role in higher ramification theory, which studies how primes split in extensions of these fields and how their valuations behave.
G. s. lang: The term 'g. s. lang' refers to a specific type of valuation in the context of ramification theory, particularly in relation to higher ramification groups. It plays a crucial role in understanding the behavior of extensions of local fields and their valuations, allowing mathematicians to explore how primes split and ramify in these extensions. This concept is pivotal for analyzing the structure of local fields and their completions.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial equation, formed by the automorphisms of a field extension that fix the base field. This concept helps us understand how different roots relate to one another and provides a powerful framework for analyzing the solvability of polynomials and the structure of number fields.
Higher ramification: Higher ramification refers to a refined concept in the study of ramification theory within algebraic number theory, focusing on the structure of ramification groups associated with a valuation in a field extension. These groups allow mathematicians to analyze how prime ideals split and behave in extensions of number fields, providing insights into the deeper relationships between number fields and their extensions.
Inertia Degree: Inertia degree is a concept in algebraic number theory that refers to the degree of the ramification of a prime ideal in an extension of number fields. It quantifies how many times a prime ideal splits, ramifies, or remains inert in the extension, helping to understand the behavior of primes across different number fields. The inertia degree specifically counts the number of primes in the extension that lie over a given prime, playing a crucial role in understanding unique factorization of ideals, ramification groups, and the overall decomposition of primes.
Kummer's Theory: Kummer's Theory is a fundamental concept in algebraic number theory that describes the behavior of ramification in the context of cyclotomic fields and how it relates to class field theory. It particularly focuses on the relationship between ramification groups, which organize the behavior of primes in extensions, and the Galois group associated with these extensions, highlighting the interplay between local and global properties of numbers.
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.
N. jacobson: n. jacobson refers to a concept in the context of ramification theory in algebraic number theory, named after Nathan Jacobson. It is used to study the behavior of ramification groups in a local field, particularly focusing on their structure and how they relate to the valuation of ideals in a number field. This concept helps in understanding the extension of fields and the intricate properties of primes within these fields.
Ramification Group: A ramification group is a concept in algebraic number theory that describes the structure of how a prime ideal in a number field splits when extended to a field extension. Specifically, these groups are used to measure how wildly or tamely a prime splits in a tower of field extensions. Understanding ramification groups is crucial for studying the behavior of primes in various extensions and provides insight into the overall structure of the number field.
Ramification Index: The ramification index is an integer that measures how a prime ideal in the base field factors into prime ideals in an extension field. It reflects the degree of 'thickness' or 'multiplicity' of the prime ideal's lift to the extension and provides insight into how the extension behaves locally. Understanding this index is crucial for exploring unique factorization of ideals, analyzing ramification groups, and examining the interaction between ramification and inertia.
Ramified Extension: A ramified extension is a type of field extension where the primes of the base field split in a specific way, particularly when at least one prime ideal in the base field ramifies in the extension. This concept is crucial for understanding how different primes behave when moving between fields and is closely tied to ramification groups that categorize these behaviors based on their level of ramification.
Tamely Ramified: Tamely ramified refers to a specific type of ramification in number theory where the ramification index is not divisible by the square of the prime in question. This means that while a prime can ramify in an extension, it does so in a controlled manner, preventing excessive complications in the structure of the extension. Understanding tamely ramified extensions is crucial for working with ramification groups and delving into higher ramification theory.
Unramified extension: An unramified extension is a type of field extension where the ramification of primes does not occur, meaning that the valuation of a prime ideal in the base field remains unchanged in the extension field. This leads to the preservation of residue fields, making it a vital concept in understanding ramification groups and higher ramification theory. Unramified extensions play a significant role in local fields, particularly in the study of their Galois groups and extensions.
Valuation ring: A valuation ring is a special type of integral domain that is associated with a valuation, which provides a way to measure the size of elements in a field. Specifically, it is a subring of a field where for any element in the field, either that element or its inverse (if it exists) belongs to the ring. This concept is crucial in understanding the structure of local fields and plays a key role in ramification theory, where it helps in analyzing how prime ideals behave under field extensions.
Wild ramification: Wild ramification refers to a type of ramification in algebraic number theory where the prime ideal's power in the factorization of a prime ideal in a field extension is greater than the usual prediction based on the degree of the extension. This situation arises especially in extensions of local fields and indicates that the inertia group associated with the ramified prime is not trivial. Understanding wild ramification is crucial for exploring deeper properties of number fields and their extensions.