1.4 Units and Dirichlet's unit theorem

Units in number fields are crucial for understanding arithmetic structure. They're invertible elements in the ring of integers, forming a multiplicative group. Their properties connect to various aspects of arithmetic geometry, including elliptic curves and Diophantine equations.

Dirichlet's unit theorem is a fundamental result describing the structure of unit groups. It states that the unit group is isomorphic to the product of roots of unity and a free abelian group of rank r, where r depends on field embeddings. This connects field geometry to arithmetic properties.

Algebraic number fields

• Fundamental structures in algebraic number theory form the foundation for studying arithmetic properties of algebraic integers
• Extend rational numbers to include roots of polynomials with rational coefficients, providing a rich framework for exploring number-theoretic problems in arithmetic geometry

Definition of number fields

Top images from around the web for Definition of number fields

• 

  General Strategy for Factoring Polynomials – Intermediate Algebra View original

  Is this image relevant?

• 

  linear algebra - Find the minimal polynomial for the given matrix. - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  abstract algebra - Cyclotomic polynomials has integer coefficients - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  General Strategy for Factoring Polynomials – Intermediate Algebra View original

  Is this image relevant?

• 

  linear algebra - Find the minimal polynomial for the given matrix. - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of number fields

• 

  General Strategy for Factoring Polynomials – Intermediate Algebra View original

  Is this image relevant?

• 

  linear algebra - Find the minimal polynomial for the given matrix. - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  abstract algebra - Cyclotomic polynomials has integer coefficients - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  General Strategy for Factoring Polynomials – Intermediate Algebra View original

  Is this image relevant?

• 

  linear algebra - Find the minimal polynomial for the given matrix. - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Finite field extensions of rational numbers $K/\mathbb{Q}$ where $K$ is algebraic over $\mathbb{Q}$
• Characterized by minimal polynomial of a primitive element $\alpha$ such that $K = \mathbb{Q}(\alpha)$
• Degree of the extension determined by the degree of the minimal polynomial
• Examples include quadratic fields ($\mathbb{Q}(\sqrt{d})$) and cyclotomic fields ($\mathbb{Q}(\zeta_n)$)

Rings of integers

• Integral closure of $\mathbb{Z}$ in a number field $K$, denoted $\mathcal{O}_K$
• Elements satisfy monic polynomials with integer coefficients
• Form a Dedekind domain with unique factorization of ideals
• Basis representation: $\mathcal{O}_K = \mathbb{Z}\omega_1 + \cdots + \mathbb{Z}\omega_n$ where $n$ is the degree of $K$
• Discriminant of the ring of integers measures arithmetic properties of the field

Units in number fields

• Play a crucial role in understanding the arithmetic structure of number fields
• Connect to various aspects of arithmetic geometry, including the study of elliptic curves and Diophantine equations

Definition of units

• Invertible elements in the ring of integers $\mathcal{O}_K$
• Characterized by norm $N_{K/\mathbb{Q}}(u) = \pm 1$ for unit $u$
• Form a multiplicative group $\mathcal{O}_K^\times$
• Examples in quadratic fields: $\mathbb{Z}[\sqrt{2}]^\times = \{\pm(1+\sqrt{2})^n : n \in \mathbb{Z}\}$

Properties of units

• Finite generation of unit group (Dirichlet's unit theorem)
• Torsion subgroup consists of roots of unity in $K$
• Units preserve ideal factorization: $u\mathfrak{a} = \mathfrak{a}$ for any ideal $\mathfrak{a}$
• Norm-one subgroup $\{u \in \mathcal{O}_K^\times : N_{K/\mathbb{Q}}(u) = 1\}$ of index 1 or 2

Dirichlet's unit theorem

• Fundamental result in algebraic number theory describing the structure of unit groups
• Connects the geometry of number fields to their arithmetic properties, essential in arithmetic geometry

Statement of the theorem

• Unit group $\mathcal{O}_K^\times$ isomorphic to $\mu_K \times \mathbb{Z}^r$ where $\mu_K$ is the group of roots of unity in $K$
• Rank $r = r_1 + r_2 - 1$ where $r_1$ is the number of real embeddings and $r_2$ is the number of pairs of complex embeddings
• Finite generation of unit group with $r$ fundamental units and torsion part $\mu_K$
• Applies to all number fields, generalizing results for quadratic fields

Geometric interpretation

• Logarithmic embedding of units into $\mathbb{R}^{r_1+r_2}$ via $\log |\sigma_i(u)|$
• Image forms a lattice of rank $r$ in a hyperplane
• Regulator measures covolume of this lattice
• Connects to geometry of numbers and Minkowski's theorem

Rank of unit group

• Determines the complexity of the unit group structure
• Influences various arithmetic properties of the number field

Free part vs torsion part

• Free part isomorphic to $\mathbb{Z}^r$, generated by fundamental units
• Torsion part $\mu_K$ consists of roots of unity, always finite
• Examples: $\mathbb{Q}(\sqrt{-1})$ has torsion $\{\pm 1, \pm i\}$ and rank 0
• Real quadratic fields have torsion $\{\pm 1\}$ and rank 1

Calculation of rank

• Formula $r = r_1 + r_2 - 1$ based on field embeddings
• $r_1$: number of real embeddings, $r_2$: number of pairs of complex embeddings
• Totality of embeddings always equals degree: $r_1 + 2r_2 = [K:\mathbb{Q}]$
• Examples: cubic fields can have ranks 0, 1, or 2 depending on their Galois group

Fundamental units

• Generate the free part of the unit group
• Essential for understanding the arithmetic of the number field

Definition and properties

• Set of $r$ units $\{\epsilon_1, \ldots, \epsilon_r\}$ generating $\mathcal{O}_K^\times / \mu_K$
• Any unit $u$ can be expressed as $u = \zeta \epsilon_1^{a_1} \cdots \epsilon_r^{a_r}$ with $\zeta \in \mu_K$ and $a_i \in \mathbb{Z}$
• Not unique, but different sets related by unimodular integer matrix
• Minimal generating set for the unit group modulo roots of unity

Methods of computation

• Minkowski's bound provides an upper limit for the norm of fundamental units
• Continued fraction algorithm for real quadratic fields
• LLL algorithm for higher degree fields
• Index calculus method for finding relations between units
• Challenges in efficiency for fields of large degree or discriminant

Regulator of number field

• Measures the "size" of the unit group
• Plays a crucial role in various number-theoretic formulas and conjectures

Definition and significance

• Determinant of the logarithmic embedding matrix of fundamental units
• Invariant of the number field, independent of choice of fundamental units
• Appears in class number formula and other important identities
• Relates to special values of L-functions (Dirichlet L-functions)

Computation techniques

• Direct computation from fundamental units via determinant
• Approximation methods using numerical computation of logarithms
• Relation to other invariants (class number, discriminant) for estimation
• Challenges in precise computation for fields of large degree

Applications of unit theorem

• Dirichlet's unit theorem finds applications in various areas of number theory and arithmetic geometry

Class number formula

• Relates class number, regulator, and other invariants of the number field
• $h_K R_K = \frac{w_K |\Delta_K|^{1/2}}{2^{r_1} (2\pi)^{r_2}} \zeta_K(1)$
• $h_K$: class number, $R_K$: regulator, $w_K$: number of roots of unity
• $\Delta_K$: discriminant, $\zeta_K(s)$: Dedekind zeta function

Pell's equation

• Diophantine equation $x^2 - dy^2 = 1$ for non-square $d$
• Solutions related to units in $\mathbb{Q}(\sqrt{d})$
• Fundamental solution corresponds to fundamental unit
• Infinite family of solutions generated by powers of fundamental unit

Generalizations

• Extensions of Dirichlet's unit theorem to broader contexts in arithmetic geometry

S-units

• Units invertible outside a finite set S of primes
• Generalization of Dirichlet's theorem: rank formula becomes $r_S = r + |S| - 1$
• Applications to S-integral points on elliptic curves
• Connection to Leopoldt's conjecture and p-adic L-functions

Function field analogue

• Unit theorem for global function fields (algebraic curves over finite fields)
• Torsion part corresponds to constant field, rank relates to genus
• Applications in coding theory and cryptography
• Connections to Drinfeld modules and t-motives

Computational aspects

• Implementation of algorithms for unit computation crucial for practical applications in arithmetic geometry

Algorithms for unit computation

• Buchmann-Lenstra algorithm for computing fundamental units
• Subexponential algorithms based on lattice reduction (LLL)
• Index calculus methods for large degree fields
• Probabilistic algorithms for approximating regulators

Software implementations

• PARI/GP provides built-in functions for unit computation
• SageMath integrates various algorithms for number field computations
• Magma offers high-performance implementations for advanced computations
• Challenges in efficiency and precision for fields of large degree or discriminant

Historical context

• Development of Dirichlet's unit theorem marks a significant milestone in the history of algebraic number theory

Development of the theorem

• Dirichlet's original proof in 1846 used analytical methods
• Minkowski's geometric number theory provided new insights
• Subsequent simplifications and generalizations by Hilbert, Hasse, and others
• Modern proofs often use adelic methods or cohomological techniques

Impact on number theory

• Foundational result for the development of algebraic number theory
• Inspired further research on class field theory and L-functions
• Connections to geometry of numbers and lattice theory
• Continuing influence on modern research in arithmetic geometry and Diophantine approximation