8.2 Dirichlet's unit theorem and its proof
4 min read•july 30, 2024
is a game-changer in algebraic number theory. It tells us exactly how many independent 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
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- Dirichlet's unit theorem provides a complete description of the structure of the group of units in the 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
- 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 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
- 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
Key Terms to Review (19)
19th-century developments: 19th-century developments refer to the significant advancements and transformations in various fields during the 1800s, particularly in mathematics and number theory. This era saw crucial innovations that laid the groundwork for modern algebra, including deeper explorations of number systems, the introduction of new theories, and an increasing emphasis on abstract concepts. These changes ultimately shaped the landscape of mathematical research and education, influencing later generations of mathematicians.
Algebraic Number Field: An algebraic number field is a finite extension of the field of rational numbers, meaning it can be constructed by adjoining roots of polynomial equations with coefficients in the rationals. This concept is fundamental in number theory, as it allows for a deeper understanding of number systems that include algebraic integers, which are roots of monic polynomials with integer coefficients. Properties such as unique factorization and the behavior of units can be studied within these fields, connecting them to broader topics like unit groups and class groups.
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 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.
Dirichlet's Theorem on Primes in Arithmetic Progressions: Dirichlet's Theorem states that in any arithmetic progression of the form $$a + nd$$, where $$a$$ and $$d$$ are coprime integers, there are infinitely many prime numbers. This result shows that primes are not just scattered among the integers but can also be found in structured patterns, which connects to deeper insights about prime distribution and the properties of numbers.
Dirichlet's Unit Theorem: Dirichlet's Unit Theorem is a fundamental result in algebraic number theory that describes the structure of the group of units in the ring of integers of a number field. It states that the unit group is isomorphic to a finite direct product of two components: a finite torsion subgroup and a free abelian group whose rank is given by the number of real embeddings minus the number of complex embeddings of the number field.
Height of a unit: The height of a unit in algebraic number theory refers to a measure that quantifies the complexity of a unit in a number field, particularly in relation to its coordinates when expressed in a certain basis. It is closely related to the concept of the units in the ring of integers of the number field and plays a crucial role in understanding the structure of the unit group. This measure provides insights into both the arithmetic and geometric properties of units, especially when applying Dirichlet's unit theorem.
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.
Link to Class Field Theory: Link to Class Field Theory refers to the connection between class field theory and the understanding of abelian extensions of number fields. It provides a framework for studying the arithmetic properties of number fields, particularly through the lens of ideal class groups and units. This concept is crucial for understanding how various algebraic structures interact, especially when examining the relationships between different number fields and their respective Galois groups.
Logarithmic embedding: Logarithmic embedding is a technique used to connect units in a number field to the real numbers through a logarithmic map, allowing for the study of the structure of the unit group. This concept is crucial in understanding the relationship between algebraic and analytic properties of units, especially in the context of Dirichlet's unit theorem, which describes the structure of the group of units in the ring of integers of a number field. By embedding units logarithmically, we can analyze their behavior and facilitate proofs regarding their properties.
Multiplicative group of units: The multiplicative group of units consists of the elements in a ring that have multiplicative inverses, meaning they can be multiplied to yield the identity element, which is usually 1. This group captures the structure of the invertible elements within a ring and is critical for understanding the behavior of units in algebraic number fields, especially in relation to Dirichlet's unit theorem.
O_k: In algebraic number theory, $$o_k$$ refers to the ring of integers of a number field $$k$$, which consists of all elements in the field that are integral over the integers. This concept is crucial as it allows mathematicians to study the algebraic properties of number fields and their relationships with the integers, providing a structured way to analyze divisibility, factorization, and units within these fields.
Peter Gustav Lejeune Dirichlet: Peter Gustav Lejeune Dirichlet was a prominent German mathematician known for his contributions to number theory and analysis in the 19th century. He is particularly noted for Dirichlet's unit theorem, which describes the structure of the group of units in the ring of integers of a number field, linking algebraic number theory to key concepts such as class numbers and unit groups.
Real quadratic field: A real quadratic field is a type of number field generated by adjoining the square root of a positive integer to the rational numbers, specifically of the form $$\mathbb{Q}(\sqrt{d})$$ where $$d$$ is a positive square-free integer. This concept connects to various important properties of algebraic integers, the structure of units, and the behavior of class groups within number theory.
Regulator: The regulator is a crucial concept in algebraic number theory that measures the size of the unit group of a number field. It captures the logarithmic growth of units and is fundamentally linked to the structure of the unit group, specifically through its role in the connection between units and class numbers. Understanding regulators helps to unveil the intricacies of Dirichlet's unit theorem, which describes the relationship between units in number fields and their ranks.
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.
Units: In number theory, units are elements of a ring that have a multiplicative inverse within that ring. This concept is crucial as it determines the structure of the number system, allowing for unique factorization and influencing the properties of algebraic structures like rings and fields. Units play a significant role in understanding how numbers behave under multiplication and how they relate to concepts like unique factorization and groups of units.
Z[k]: In the context of algebraic number theory, z[k] refers to the ring of integers of a number field K, where k is a specific algebraic number. This ring contains all the algebraic integers that can be expressed as elements in the field generated by k over the rational numbers. Understanding z[k] is essential as it lays the foundation for exploring properties of ideals, units, and class groups in algebraic number theory.