Norm and trace 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

• Norm of element α in number field K of degree n over Q defined as product of all conjugates of α
  • Denoted as $N(α) = \prod σ(α)$
  • 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 $Tr(α) = \sum σ(α)$
  • 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 field extension 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: $N(a + b\sqrt{d}) = a^2 - db^2$
  • Trace of a + b√d: $Tr(a + b\sqrt{d}) = 2a$
• 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: $N(αβ) = N(α)N(β)$ for any α, β in K
  • Proved using definition and properties of field embeddings
• Trace additivity: $Tr(α + β) = Tr(α) + Tr(β)$ for any α, β in K
  • Follows from linearity of field embeddings
• Scalar multiplication properties (r rational, α in K, n = [K:Q])
  • $N(rα) = r^n N(α)$
  • $Tr(rα) = r Tr(α)$
• Norm of unit in ring of integers of K always ±1
  • Crucial for studying unit group of number fields
• Trace of algebraic integer 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