Dirichlet's unit theorem is a game-changer in algebraic number theory. It tells us exactly how many independent units exist in a number field, connecting this to the field's real and complex embeddings.
This theorem is crucial for understanding the structure of number fields. It helps solve equations, study ideal class groups, and lays the groundwork for more advanced topics in the field.
Dirichlet's Unit Theorem and its Implications
Structure and Statement of the Theorem
- Dirichlet's unit theorem provides a complete description of the structure of the group of units in the ring of integers of a number field
- For a number field K of degree n over Q, with r1 real embeddings and r2 pairs of complex embeddings, the group of units in the ring of integers OK is isomorphic to the direct product of the group of roots of unity in K and a free abelian group of rank r = r1 + r2 - 1
- The rank r is also known as the unit rank of the number field K
- Group of units in OK is finitely generated
- Generalizes the structure of units in quadratic number fields (real and complex) to arbitrary number fields
- Real quadratic fields have infinite units (fundamental unit and its powers)
- Complex quadratic fields have finite units (roots of unity)
Implications and Applications
- Significant implications for the study of ideal class groups
- Connects to the finiteness of the class number
- Crucial for solving Diophantine equations over number fields
- Helps in finding integral points on elliptic curves
- Essential for advanced topics in algebraic number theory
- Class field theory (description of abelian extensions)
- Study of L-functions (generalizations of the Riemann zeta function)
- Provides a framework for computational number theory
- Algorithms for finding fundamental units
- Methods for solving norm equations
Proving Dirichlet's Unit Theorem
Logarithmic Embedding and Lattice Structure
- Define the logarithmic embedding of the group of units into a real vector space
- Maps units to vectors using logarithms of absolute values of embeddings
- Prove the image of the logarithmic embedding is a lattice in a hyperplane of the vector space
- Hyperplane defined by the sum of coordinates being zero
- Demonstrate the kernel of the logarithmic embedding consists only of the roots of unity in the number field
- Uses properties of complex absolute value
- Establish the connection between the dimension of the lattice and the number of real and complex embeddings of the number field
- Dimension is r = r1 + r2 - 1
Fundamental Domain and Finiteness
- Use the pigeonhole principle to show there exist units with arbitrarily small logarithmic embeddings
- Divides a large region into smaller cells
- Construct a fundamental domain for the action of the units on the upper half-plane (for real quadratic fields) or its higher-dimensional analogs
- Generalizes to n-dimensional hyperbolic space for higher degree fields
- Apply the theory of continued fractions (for real quadratic fields) or its generalizations to show the fundamental domain has finite volume
- Uses properties of hyperbolic geometry
- Demonstrate the finiteness of the volume implies the group of units is finitely generated
- Connects geometric and algebraic properties
Logarithmic Embedding for Units
Definition and Properties
- Define the logarithmic embedding as a map from the group of units to a real vector space of dimension r1 + r2
- For a unit ε, L(ε) = (log|σ1(ε)|, ..., log|σr1+r2(ε)|)
- Logarithmic embedding transforms multiplication of units into addition of vectors
- L(εη) = L(ε) + L(η) for units ε and η
- Image of the logarithmic embedding lies in a hyperplane defined by the sum of coordinates being zero
- Consequence of the product formula for number fields
- Logarithmic embedding is injective modulo roots of unity
- If L(ε) = L(η), then ε/η is a root of unity
Applications and Computational Aspects
- Logarithmic embedding defines a notion of size for units, independent of their algebraic representation
- ||L(ε)|| measures the "size" of the unit ε
- Study of units through logarithmic embeddings relates to Minkowski's geometry of numbers
- Connects to lattice theory and convex geometry
- Logarithmic embedding used to approximate fundamental units computationally
- LLL algorithm for lattice basis reduction
- Allows visualization of unit groups for low-degree number fields
- Plots in 2D or 3D space for quadratic and cubic fields
The Regulator's Role in the Proof
Definition and Significance
- Regulator defined as the volume of the fundamental parallelotope of the lattice formed by the logarithmic embedding of a system of fundamental units
- R = |det(L(ε1), ..., L(εr))| where ε1, ..., εr are fundamental units
- Regulator measures the "density" of units in the number field
- Larger regulator indicates sparser distribution of units
- Non-zero regulator crucial for proving the group of units modulo roots of unity is free of rank r
- Ensures the fundamental units are linearly independent over Z
Connections to Other Number Field Invariants
- Regulator appears in the residue of the Dedekind zeta function at s = 1
- Connects to class number and other important invariants
- Bounds on the regulator used to estimate the size of fundamental units
- Helps in computational aspects of finding units
- Relationship between the regulator and the class number formula for number fields
- R * h = |μ| * √|D| * res(ζK, 1) / (2r1+r2 * π r2)
- Regulator computation significant in algorithmic number theory
- Used in algorithms for computing class groups and fundamental units