Glossary

4.1 Norm and trace maps for field extensions

4 min read•july 30, 2024

and maps are essential tools in algebraic number theory, connecting field extensions to their base fields. These maps provide a way to analyze the structure and properties of field extensions, offering insights into their arithmetic and algebraic characteristics.

Understanding norm and trace maps is crucial for grasping the broader concepts of norms, traces, and discriminants in field theory. These functions help us explore the relationships between different number fields and play a vital role in studying algebraic structures and their applications.

Norm and Trace Maps

Definitions and Basic Properties

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  • Norm map N: L → K for finite field extension L/K defined as determinant of K-linear transformation of multiplication by α in L
  • Trace map Tr: L → K for finite field extension L/K defined as sum of diagonal elements of matrix representation of multiplication by α in L
  • Both maps are K-linear functions mapping elements from L to K
  • For separable extension L/K of degree n, norm and trace expressed using conjugates of α
    • N(α) = ∏σ(α)\prod \sigma(α) where σ ranges over all K-embeddings of L into its algebraic closure
    • Tr(α) = ∑σ(α)\sum \sigma(α) where σ ranges over all K-embeddings of L into its algebraic closure
  • Norm map multiplicative N(αβ)=N(α)N(β)N(αβ) = N(α)N(β) for all α, β in L
  • Trace map additive Tr(α+β)=Tr(α)+Tr(β)Tr(α + β) = Tr(α) + Tr(β) for all α, β in L
  • For simple L = K(α), norm and trace computed using of α over K

Galois Theory and Field Structure

  • Norm and trace maps are Galois-equivariant commuting with action of Galois group of extension
  • Image of norm map subgroup of multiplicative group K*
  • Image of trace map additive subgroup of K
  • Fixed field of kernel of norm (or trace) map corresponds to important subfields of extension
  • Behavior of norm and trace maps under composition reflects tower law for degrees of field extensions
  • Norm map extends notion of absolute value from base field to extension field (crucial for local fields and p-adic numbers)
  • Trace forms defined using trace map significant in understanding field extension structure (separability and ramification theory)

Computing Norms and Traces

Methods for Simple Extensions

  • For L = K(α) with minimal polynomial f(x)=anxn+an−1xn−1+...+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0
    • Norm of α: N(α)=(−1)na0N(α) = (-1)^n a_0
    • Trace of α: Tr(α)=−an−1Tr(α) = -a_{n-1}
  • Finite fields use Frobenius automorphism and its powers for computation
  • Matrix methods employ matrix representation of multiplication map
    • Calculate determinant for norm
    • Calculate trace for trace
  • Simplified formulas for special cases (quadratic or cyclotomic extensions)
    • Example (quadratic extension): For L=K(d)L = K(\sqrt{d}), N(a+bd)=a2−db2N(a + b\sqrt{d}) = a^2 - db^2 and Tr(a+bd)=2aTr(a + b\sqrt{d}) = 2a

Advanced Techniques

  • Composite field extensions L/M/K use transitivity property
    • NL/K=NM/K∘NL/MN_{L/K} = N_{M/K} ∘ N_{L/M}
    • TrL/K=TrM/K∘TrL/MTr_{L/K} = Tr_{M/K} ∘ Tr_{L/M}
  • Computational techniques for complex field extensions
    • Use primitive elements
    • Employ basis representations
    • Apply Galois theory to simplify calculations
  • Example (cyclotomic extension): For L=K(ζn)L = K(\zeta_n) where ζn\zeta_n is a primitive nth root of unity
    • N(ζn)=1N(\zeta_n) = 1 if n is odd, N(ζn)=−1N(\zeta_n) = -1 if n is even
    • Tr(ζn)=0Tr(\zeta_n) = 0 if n > 1

Properties of Norm and Trace Maps

Algebraic Properties

  • Norm map multiplicative N(αβ)=N(α)N(β)N(αβ) = N(α)N(β) for all α, β in L
  • Trace map additive Tr(α+β)=Tr(α)+Tr(β)Tr(α + β) = Tr(α) + Tr(β) for all α, β in L
  • Both maps K-linear respecting scalar multiplication and addition over base field K
  • Norm of element non-zero if and only if element non-zero (useful for determining invertibility)
  • Trace of element zero for all elements in L if and only if characteristic of K divides [L:K]
  • Example: In Q(2)\mathbb{Q}(\sqrt{2}) over Q\mathbb{Q}
    • N(a+b2)=a2−2b2N(a + b\sqrt{2}) = a^2 - 2b^2
    • Tr(a+b2)=2aTr(a + b\sqrt{2}) = 2a

Field Extension Structure

  • Norm and trace maps provide connection between algebraic structure and arithmetic properties of field extensions
  • Norm map used to extend absolute value notion from base field to extension field
  • Trace forms play role in understanding field extension structure (separability and ramification theory)
  • Fundamental in study of relative discriminants and different ideals (important invariants in algebraic number theory)
  • Essential tools in local and global class field theory relating arithmetic of extensions to Galois groups
  • Example: In cyclotomic extension Q(ζp)\mathbb{Q}(\zeta_p) over Q\mathbb{Q} where ζp\zeta_p is a primitive pth root of unity
    • Norm map relates to cyclotomic units
    • Trace map connects to Gauss sums

Norm and Trace Maps vs Field Extensions

Applications in Algebraic Number Theory

  • Norm and trace maps crucial in studying arithmetic of algebraic number fields
  • Used to define and study important invariants (discriminants, different ideals)
  • Play role in formulating and proving reciprocity laws in class field theory
  • Help characterize ramification behavior in extensions of local and global fields
  • Example: In quadratic field Q(d)\mathbb{Q}(\sqrt{d})
    • Norm of ideal relates to factorization of primes
    • Trace form determines whether extension is tamely or wildly ramified

Connections to Other Areas of Mathematics

  • Norm and trace maps bridge algebraic number theory with other mathematical disciplines
  • Used in constructing and analyzing central simple algebras and division algebras
  • Apply to representation theory through character values and Frobenius reciprocity
  • Appear in algebraic geometry in the context of étale cohomology and Grothendieck trace formula
  • Utilized in coding theory for constructing and analyzing certain error-correcting codes
  • Example: Norm map in quaternion algebras
    • Defines reduced norm used in studying arithmetic of quaternions
    • Connects to theory of quadratic forms and special linear groups

Key Terms to Review (18)

Absolute Norm: The absolute norm is a concept in algebraic number theory that refers to a specific way of measuring the size of elements in a number field. It is defined as the product of all the embeddings of an algebraic number into the complex numbers, giving insight into its properties and relationships within the field. This norm helps in understanding field extensions, as it provides a way to relate elements across different fields, revealing structural aspects that are crucial for further analysis.
Additivity of Trace: The additivity of trace is a property of the trace function defined for field extensions that states if you have a finite extension of fields, the trace of the sum of two elements is equal to the sum of their traces. This property highlights the linear nature of the trace function and its connection to the structure of field extensions, allowing for simplifications in calculations involving traces.
Algebraic extension: An algebraic extension is a type of field extension where every element of the extended field is a root of a non-zero polynomial with coefficients from the base field. This means that when you have a field and you create a new field by adding elements that satisfy certain polynomial equations, you end up with an algebraic extension. These extensions play a key role in understanding the structure of fields, especially when discussing concepts like algebraic closures and how different field extensions relate to one another.
Algebraic Integers: Algebraic integers are complex numbers that are roots of monic polynomials with integer coefficients. They play a crucial role in number theory, particularly in the study of unique factorization and properties of number fields, connecting various concepts like integral bases and prime ideals.
Characteristic Polynomial: The characteristic polynomial is a polynomial associated with a square matrix or a linear transformation that encodes important information about its eigenvalues. This polynomial is formed by taking the determinant of the matrix subtracted by a scalar multiple of the identity matrix, typically expressed as $$P(t) = ext{det}(A - tI)$$, where $A$ is the matrix and $t$ represents the eigenvalue. It reveals the roots, which correspond to the eigenvalues, providing insights into the structure and properties of the matrix.
Cohomology of Field Extensions: Cohomology of field extensions is a mathematical concept that studies the relationship between algebraic structures and their cohomological properties. It provides a framework to analyze how certain algebraic invariants behave under field extensions, particularly through the lens of norm and trace maps. This notion is important in understanding the interaction between cohomology groups and various types of field extensions, enriching the study of algebraic number theory.
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.
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.
Integral Closure: Integral closure refers to the set of all elements in a given field that are integral over a specified ring, particularly focusing on algebraic integers. It connects various concepts like algebraic numbers and integers, providing a way to understand the structure of rings of integers in number fields, ensuring that algebraic properties are preserved within extensions.
Isomorphism of Fields: An isomorphism of fields is a bijective homomorphism between two field structures, meaning that it preserves both the operations of addition and multiplication. This concept establishes a strong relationship between two fields, showing they are structurally the same even if they are represented differently. Understanding isomorphisms helps in analyzing the properties of field extensions, particularly through the lens of norms and trace maps, which provide insights into how elements relate within different field contexts.
Minimal Polynomial: The minimal polynomial of an algebraic element over a field is the monic polynomial of smallest degree that has the element as a root. This polynomial captures the essence of the element's algebraic properties and relates closely to the structure of number fields, field extensions, and their algebraic closures.
Multiplicative Property of Norm: The multiplicative property of norm states that for any two elements in a number field, the norm of their product is equal to the product of their norms. This property is essential in understanding how norms interact within field extensions and helps in simplifying calculations involving norms when working with algebraic integers.
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.
Norm of an Ideal: The norm of an ideal is a fundamental concept in algebraic number theory that measures the 'size' of an ideal within a ring of integers in a number field. Specifically, it is defined as the index of the ideal in the ring, giving a way to quantify how many elements in the ring can be represented as multiples of generators of that ideal. This notion is crucial for understanding various properties related to field extensions and prime factorization within these structures.
Norm-trace theorem: The norm-trace theorem is a fundamental result in algebraic number theory that relates the norm and trace of elements in a field extension. It states that for a finite field extension, the norm of an element is equal to the product of its conjugates, while the trace is the sum of its conjugates. This theorem provides essential insights into the arithmetic properties of field extensions and is crucial for understanding Galois theory and related concepts.
Relative Trace: The relative trace is a function that, given a field extension, takes an element from the larger field and returns a sum of its conjugates over a smaller field. This concept is essential in understanding how elements behave under field extensions and plays a crucial role in the study of algebraic integers and their properties.
Trace: In the context of algebraic number theory, the trace of an element in a number field is the sum of the conjugates of that element over its minimal polynomial. It plays a crucial role in understanding the structure of number fields, as it relates to important concepts like the norm and characterizes properties such as the behavior of field extensions and algebraic integers.
Trace of an algebraic element: The trace of an algebraic element is the sum of the conjugates of that element within a given field extension. This concept arises when considering field extensions, as the trace provides insights into the algebraic structure of the field and its elements, relating closely to norms and other mappings. Understanding the trace is crucial for studying properties such as Galois groups and integral closures in algebraic number theory.
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