2.3 Norm and trace in number fields
4 min read•july 30, 2024
and are powerful tools in number fields, helping us understand the arithmetic of algebraic numbers. They generalize familiar concepts from linear algebra, providing a bridge between abstract algebra and number theory.
These functions map elements of a number field to rational numbers, capturing essential information about field embeddings and algebraic properties. Norm and trace play crucial roles in factorizing ideals, solving Diophantine equations, and exploring the structure of rings of integers.
Norm and trace of elements
Definition and basic concepts
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- Norm of element α in number field K of degree n over Q defined as product of all conjugates of α
- Denoted as
- σ ranges over all embeddings of K into C
- Trace of element α in number field K of degree n over Q defined as sum of all conjugates of α
- Denoted as
- σ ranges over all embeddings of K into C
- Both norm and trace map elements of K to rational numbers (functions from K to Q)
- Expressed as determinant and trace of matrix representing multiplication by α in K/Q
- For primitive element α of K, computed using minimal polynomial of α over Q
- Algebraic invariants providing important information about arithmetic properties of elements in number fields
- Generalize familiar notions of determinant and trace from linear algebra to field extensions
Relationship to field embeddings
- Norm and trace defined using field embeddings (homomorphisms from K to C fixing Q)
- Number of distinct embeddings equals degree of field extension [K:Q]
- For real embeddings, conjugates are real numbers
- For complex embeddings, conjugates occur in complex conjugate pairs
- Galois theory connects embeddings to automorphisms of K fixing Q
- Understanding embeddings crucial for computing and interpreting norm and trace values
Calculating norm and trace
Explicit formulas for specific number fields
- Quadratic number fields Q(√d), d square-free integer
- Norm of a + b√d:
- Trace of a + b√d:
- Cubic number fields
- Use cubic formula and Vieta's formulas relating roots to coefficients
- Example: For x^3 + px + q = 0, norm of root α is -q, trace is 0
- Cyclotomic fields Q(ζn), ζn primitive nth root of unity
- Utilize properties of cyclotomic polynomials
- Example: In Q(ζ5), N(ζ5) = 1, Tr(ζ5) = -1
General computational techniques
- Find minimal polynomial of element and use its coefficients
- Employ resultants and discriminants for efficient calculations in higher degree fields
- Utilize Galois group of number field to simplify calculations for certain elements
- Develop proficiency through practice with various examples across different types of number fields
- Use computer algebra systems (Sage, PARI/GP) for complex calculations
- Apply matrix methods for norm and trace computations in higher dimensional extensions
Properties of norm and trace
Algebraic properties
- Norm multiplicativity: for any α, β in K
- Proved using definition and properties of field embeddings
- Trace additivity: for any α, β in K
- Follows from linearity of field embeddings
- Scalar multiplication properties (r rational, α in K, n = [K:Q])
- Norm of unit in ring of integers of K always ±1
- Crucial for studying unit group of number fields
- Trace of always rational integer
- Important for structure of rings of integers
- Galois-invariance of norm and trace
- Unchanged under action of any automorphism of K fixing Q
Advanced properties and relationships
- Norm and trace behavior under field extensions
- Connection to other algebraic invariants (discriminants, different)
- Relationship between norm, trace, and minimal polynomial coefficients
- Norm and trace in composite field extensions
- Hilbert's Theorem 90 and its connection to norm
- Local-global principles for norms in number fields
Applications of norm and trace
Number-theoretic applications
- Factorization of ideals in ring of integers of number field
- Crucial for understanding ideal class group
- Solving Diophantine equations
- Use multiplicativity of norm to reduce to equations over integers
- Example: Solving x^2 - dy^2 = ±1 (Pell's equation) using norms in Q(√d)
- Study of trace forms (quadratic forms derived from trace)
- Used to investigate arithmetic of number fields and subfields
- Defining and studying size measures for algebraic numbers (Mahler measure)
- Local field theory
- Analogous definitions of norm and trace
- Used to study ramification and other local properties
Practical and advanced applications
- Integral bases and discriminants of number fields
- Essential for understanding arithmetic structure
- Cryptography
- Construction of public-key cryptosystems based on algebraic number theory
- Example: NTRU cryptosystem using norms in certain number fields
- Class field theory
- Norm residue symbols and their connection to reciprocity laws
- Algebraic geometry
- Norm and trace in function fields and their geometric interpretations
- Representation theory
- Character theory of finite groups using traces in number fields
Key Terms to Review (16)
Additive Property of Traces: The additive property of traces refers to the fundamental principle that the trace of the sum of two elements in a number field is equal to the sum of their traces. This property illustrates how the trace function behaves in relation to addition, and it plays a vital role in understanding the relationships between elements in algebraic number theory, particularly in the study of number fields and their extensions.
Algebraic Conjugates: Algebraic conjugates are the roots of a polynomial equation with rational coefficients that share the same minimal polynomial. They play a crucial role in understanding the structure of number fields, particularly in relation to the norm and trace functions. By exploring these conjugates, we can better analyze the behavior of algebraic numbers and their relationships within a number field.
Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. These numbers include all integers and roots of unity, forming a crucial part of algebraic number theory, particularly when discussing the properties of number fields, norms, traces, minimal polynomials, and the structure of Dedekind domains.
Class number: The class number is an important invariant in algebraic number theory that measures the failure of unique factorization in the ring of integers of a number field. It reflects how many distinct ideal classes exist, indicating whether every ideal can be expressed uniquely as a product of prime ideals. A class number of one means that unique factorization holds, while a higher class number suggests complications in the structure of ideals within the number field.
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 Theorem: Dedekind's Theorem states that in a number field, the ideal class group is finitely generated and provides a fundamental connection between the arithmetic of number fields and the properties of their ideals. This theorem shows how ideals can be classified and how their structure relates to the field's algebraic properties, particularly through the norm and trace functions, as well as their behavior under ramification and inertia in extensions of number fields.
Emmy Noether: Emmy Noether was a groundbreaking mathematician whose work significantly shaped modern algebra and number theory. She is best known for Noether's Theorem, which establishes a deep connection between symmetry and conservation laws in physics, but her contributions extend into the realm of algebraic integers and number fields, where she introduced innovative ideas about integral bases and ideal arithmetic. Her insights laid the foundation for many concepts that are essential for understanding advanced topics in algebra and number theory.
Factorization in number fields: Factorization in number fields refers to the process of expressing an element of a number field as a product of irreducible elements or primes within that field. This concept is crucial as it extends the classical notion of factorization from integers to more complex structures, allowing for a richer understanding of arithmetic properties. Factorization plays an important role in the study of the norm and trace, as it helps to analyze how these concepts interact with ideals and their decomposition in different number fields.
Field Extension: A field extension is a bigger field that contains a smaller field and allows for more solutions to polynomial equations. This concept helps in understanding how different fields relate to each other, especially when it comes to algebraic numbers, algebraic integers, and the properties of polynomials in those fields.
Finite Field: A finite field, also known as a Galois field, is a set equipped with two operations, addition and multiplication, satisfying the properties of a field, but containing a finite number of elements. This concept is fundamental in various mathematical disciplines, including algebraic structures where fields play a critical role, as well as in number theory and applications in coding theory and cryptography.
Multiplicative Property of Norms: The multiplicative property of norms states that for any two elements in a number field, the norm of their product equals the product of their norms. This property is crucial when working with ideals and understanding the structure of number fields, especially when analyzing how norms interact with various algebraic properties like divisibility and factorization.
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 a prime ideal: The norm of a prime ideal in a number field is a fundamental concept that represents the size or measure of the ideal. It is defined as the index of the ideal in the ring of integers of the number field, essentially quantifying how many elements from the ring can be generated by the ideal. This idea is closely connected to understanding the behavior of prime ideals within the context of algebraic number theory, particularly when analyzing how these ideals interact with elements of the field and their factorization 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 Formula: The trace formula is a mathematical expression that connects the trace of an algebraic integer and its conjugates within a number field. It plays a critical role in understanding the structure of number fields by linking the properties of their elements with their algebraic characteristics. This relationship allows us to calculate the trace of an element as the sum of its Galois conjugates, which helps in exploring important concepts like norms and discriminants.
Trace of an algebraic integer: The trace of an algebraic integer is the sum of its conjugates, which are the roots of its minimal polynomial over the integers. This concept is crucial in understanding the structure of number fields, as it relates to the behavior and properties of algebraic integers. The trace can provide insights into how elements within a number field interact with one another, especially in terms of their relationships to other algebraic integers and the overall field structure.