4.3 Relation between discriminants and field extensions
4 min read•july 30, 2024
Discriminants are powerful tools in algebraic number theory, revealing crucial information about field extensions. They encode arithmetic properties, behavior, and field structure, serving as a key to understanding the complexity of number fields.
Exploring discriminants connects various concepts in the chapter on norms, traces, and discriminants. From basic calculations to advanced applications in field classification, discriminants provide insights into the intricate relationships between different number fields and their properties.
Discriminants for field extensions
Fundamental properties of discriminants
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- provides information about the structure of its splitting field
- Number field K defined as discriminant of minimal polynomial of primitive element over Q
- Integer that encodes arithmetic properties of the field and its
- Absolute value relates to volume of fundamental domain of ring of integers (provides measure of complexity of algebraic integers)
- Sign reveals number of real and complex embeddings (positive for totally real fields, negative for fields with complex embeddings)
- Prime factorization exposes ramification in field extension (each ramified prime divides the discriminant)
- Invariant under isomorphism (isomorphic number fields share same discriminant)
Mathematical formulation and calculations
- Field extension K/F discriminant defined as determinant of trace form matrix for basis of K over F
- Non-zero discriminant indicates separable extension K/F
- Galois extensions relate discriminant to Galois group order and different ideal
- Compositum of field extensions expresses discriminant using individual extension discriminants and relative discriminants
- p-adic valuation of discriminant reveals ramification index and inertia degree of prime ideals above p
- Quadratic extensions Q(√d)/Q have discriminant 4d if d ≡ 2,3 (mod 4) and d if d ≡ 1 (mod 4)
- Determines if polynomial defines number field with specific properties (totally real, particular Galois group)
Discriminants and ramification
Relationship between discriminants and prime factorization
- Number field discriminant divisible by ramifying primes in field extension
- Prime factor exponent in discriminant relates to ramification indices of prime ideals above it
- Galois extension K/Q with group G has discriminant form ±∏p^(e_p-1), e_p as ramification index of p
- Conductor-discriminant formula connects abelian extension discriminant to conductor (measures ramification)
- Wild ramification occurs when base field characteristic divides ramification index, increasing discriminant exponents
- Different ideal measures local ramification at each prime ideal (closely related to discriminant)
- Dedekind's discriminant theorem states prime p ramifies in number field K if and only if p divides discriminant of K
Advanced concepts in discriminants and ramification
- p-adic valuation of discriminant provides detailed information on ramification structure
- Higher p-adic valuation indicates more severe ramification
- Can be used to determine tamely and wildly ramified primes
- Discriminant factors as product of local discriminants at each prime
- Local discriminants relate to ramification and inertia at specific primes
- Artin conductor refines conductor-discriminant formula for non-abelian extensions
- Involves characters of Galois representations
- Discriminants of relative extensions K/L relate to absolute discriminants of K and L
- Formula:
- Higher ramification groups provide finer analysis of wild ramification
- Contribute to more complex discriminant formulas in wildly ramified cases
Classifying number fields with discriminants
Applications in number field classification
- Bounds degree of number fields with given properties (class number one fields)
- Crucial for determining integral basis of number field (essential for studying arithmetic properties)
- Key invariant in classification of number fields (particularly for small degree fields)
- Minkowski's bound involves discriminant, limits absolute value of discriminant for fields of given degree and signature
- Studies distribution of number fields (Davenport-Heilbronn theorem on density of cubic fields)
- Q(ζ_n) have discriminant given by simple formula with conductor n and Euler totient function φ(n)
- Behavior under field extensions used to study towers of number fields and asymptotic properties
Advanced techniques and theorems
- Odlyzko bounds provide tighter estimates for discriminants of number fields
- Improve upon Minkowski's bound using analytic techniques
- Hermite's theorem relates discriminants to field degrees
- Finite number of number fields with given discriminant
- Hilbert class fields have discriminants related to class numbers of base fields
- where H is Hilbert class field of K and h_K is class number of K
- Stark's conjectures connect special values of L-functions to discriminants and regulators
- Provide deep insights into arithmetic of number fields
- Discriminant product formula for normal extensions:
- where d_p depends on ramification groups
- Genus theory uses discriminants to study ideal class groups
- Particularly effective for
Key Terms to Review (15)
Cyclotomic Fields: Cyclotomic fields are number fields generated by adjoining a primitive root of unity, typically denoted as $$\ ext{zeta}_n = e^{2\pi i / n}$$, where $$n$$ is a positive integer. These fields are significant in number theory, particularly in studying the properties of algebraic integers, Galois groups, and class numbers. Cyclotomic fields reveal deep connections between algebra and geometry through their roots of unity, which also impact discriminants and their calculations, field extensions, and ramification behavior.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dedekind's Criterion: Dedekind's Criterion provides a way to determine whether a given number field is a Dedekind domain by examining the factorization of ideals in its ring of integers. This criterion connects algebraic integers, number fields, and the behavior of prime ideals within those fields, highlighting the relationship between algebraic structures and their integral bases.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Discriminant of a number field: The discriminant of a number field is a crucial invariant that provides information about the arithmetic properties of the field, specifically related to its algebraic integers. It can be defined in terms of the minimal polynomial of a generator of the field, and it plays an essential role in determining the structure of the field's ring of integers. The discriminant is intimately connected with concepts like ramification in extensions, class numbers, and the behavior of ideals.
Discriminant of a polynomial: The discriminant of a polynomial is a mathematical expression that provides important information about the roots of the polynomial, specifically the nature and multiplicity of these roots. It helps determine whether the roots are real or complex, and whether they are distinct or repeated. In the context of field extensions, the discriminant reveals relationships between number fields and can indicate the ramification of primes in those fields.
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.
Ideal Class Group: The ideal class group is a fundamental concept in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It consists of equivalence classes of fractional ideals, where two ideals are considered equivalent if their product with a principal ideal is again a fractional ideal. This group plays a crucial role in understanding the structure of rings of integers and their relationship to number fields, helping to connect various areas such as discriminants, integral bases, and the properties of Dedekind domains.
Kronecker-Weber Theorem: The Kronecker-Weber Theorem states that every abelian extension of the rational numbers can be obtained by adjoining a root of unity and a root of a cyclotomic field. This theorem is significant because it provides a comprehensive understanding of how certain field extensions relate to number fields, particularly in the context of their structure and properties.
Norm: In algebraic number theory, the norm of an algebraic number is a value that gives important information about its behavior in relation to a field extension. It can be viewed as a multiplicative measure that reflects how the number scales when considered within its minimal field, connecting properties of elements with their corresponding fields and extensions.
Quadratic fields: Quadratic fields are number fields that can be expressed in the form $\mathbb{Q}(\sqrt{d})$, where $d$ is a square-free integer. These fields are significant because they provide a rich structure for studying properties of numbers, including their ring of integers, discriminants, and class numbers, all of which relate to broader concepts in number theory.
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 Dedekind: Richard Dedekind was a prominent German mathematician known for his contributions to abstract algebra and number theory, particularly in the development of ideals and the concept of Dedekind domains. His work laid the foundation for understanding the structure of number fields and their properties, which are central to modern algebraic number theory.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Square-free: A number is called square-free if it is not divisible by the square of any prime number. This property is important when studying discriminants and field extensions, as square-free numbers help to characterize the nature of roots and their relationships in algebraic structures. In particular, square-free discriminants can indicate whether the corresponding polynomial has distinct roots, which affects the behavior of field extensions derived from those polynomials.