🔢Algebraic Number Theory Unit 8 – Dirichlet's Unit Theorem

Key Concepts and Definitions

• Algebraic number field $K$ finite extension of the rational numbers $\mathbb{Q}$
• Ring of integers $\mathcal{O}_K$ consists of elements in $K$ that are roots of monic polynomials with integer coefficients
• Unit group $U_K$ multiplicative group of invertible elements in $\mathcal{O}_K$
  • Units have norm $\pm 1$ and form a group under multiplication
• Rank of the unit group $r = r_1 + r_2 - 1$, where $r_1$ is the number of real embeddings and $r_2$ is the number of pairs of complex embeddings of $K$
• Fundamental unit generates a subgroup of finite index in $U_K$
• Regulator $R_K$ logarithmic volume of the fundamental domain of the unit group
• Dirichlet's Unit Theorem describes the structure of the unit group $U_K$ in terms of its rank and torsion subgroup

Historical Context and Development

• Dirichlet's Unit Theorem first proved by Peter Gustav Lejeune Dirichlet in 1846
• Builds upon earlier work by Lagrange and Gauss on quadratic forms and binary quadratic forms
• Dirichlet introduced the logarithmic embedding of the unit group, enabling a geometric interpretation
• Subsequent generalizations and refinements by Dedekind, Minkowski, and others
  • Dedekind extended the theorem to ideals and introduced the concept of the regulator
• Fundamental result in algebraic number theory, laying the foundation for the study of unit groups and class groups
• Connections to the Dedekind zeta function and the analytic class number formula discovered later

Statement of Dirichlet's Unit Theorem

• Let $K$ be an algebraic number field with ring of integers $\mathcal{O}_K$ and unit group $U_K$
• The unit group $U_K$ is finitely generated and has the structure $U_K \cong \mu_K \times \mathbb{Z}^{r_1+r_2-1}$, where:
  • $\mu_K$ is the torsion subgroup consisting of roots of unity in $K$
  • $r_1$ is the number of real embeddings of $K$
  • $r_2$ is the number of pairs of complex embeddings of $K$
• The rank of the unit group is $r = r_1 + r_2 - 1$
• There exist fundamental units $\varepsilon_1, \ldots, \varepsilon_r$ such that every unit $u \in U_K$ can be uniquely written as $u = \zeta \varepsilon_1^{n_1} \cdots \varepsilon_r^{n_r}$, where $\zeta \in \mu_K$ and $n_1, \ldots, n_r \in \mathbb{Z}$

Proof Outline and Key Steps

• Consider the logarithmic embedding $\ell: U_K \rightarrow \mathbb{R}^{r_1+r_2}$ given by $\ell(u) = (\log |\sigma_1(u)|, \ldots, \log |\sigma_{r_1+r_2}(u)|)$, where $\sigma_1, \ldots, \sigma_{r_1+r_2}$ are the embeddings of $K$
• Show that the image $\ell(U_K)$ is a discrete subgroup of the hyperplane $H = \{(x_1, \ldots, x_{r_1+r_2}) \in \mathbb{R}^{r_1+r_2} : \sum_{i=1}^{r_1+r_2} x_i = 0\}$
• Prove that the kernel of $\ell$ is precisely the torsion subgroup $\mu_K$
• Use Minkowski's theorem on lattice points to show that $\ell(U_K)$ has rank $r = r_1 + r_2 - 1$
• Conclude that $U_K \cong \mu_K \times \mathbb{Z}^{r_1+r_2-1}$ and choose fundamental units corresponding to a basis of $\ell(U_K)$

Applications in Algebraic Number Theory

• Dirichlet's Unit Theorem plays a crucial role in the study of the ideal class group and the class number of an algebraic number field
• Used in the proof of the finiteness of the ideal class group and the Dirichlet-Chevalley-Hasse unit theorem
• Fundamental in the study of the distribution of prime ideals and the Chebotarev density theorem
• Connections to the Dedekind zeta function and the analytic class number formula
  • The residue of the Dedekind zeta function at $s=1$ is related to the regulator and the class number
• Applications in the study of Diophantine equations and the unit equation
• Generalizations to S-unit groups and S-class groups in the context of arithmetic geometry

Examples and Illustrations

• In the quadratic field $K = \mathbb{Q}(\sqrt{2})$, the unit group is generated by $-1$ and the fundamental unit $1 + \sqrt{2}$
• For the cyclotomic field $K = \mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n$-th root of unity, the unit group has rank $\varphi(n)/2 - 1$, where $\varphi$ is Euler's totient function
• In the cubic field $K = \mathbb{Q}(\sqrt[3]{2})$, the unit group is generated by $-1$ and the fundamental unit $1 + \sqrt[3]{2}$
• The Gaussian integers $\mathbb{Z}[i]$ have unit group $\{\pm 1, \pm i\}$, consisting only of torsion units
• The Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega = (-1 + \sqrt{-3})/2$, have unit group $\{\pm 1, \pm \omega, \pm \omega^2\}$

Related Theorems and Connections

• Dirichlet's Unit Theorem is a generalization of the Dirichlet-Chevalley-Hasse unit theorem for S-unit groups
• Closely related to the finiteness of the ideal class group and the Dirichlet-Chevalley-Hasse class number formula
• Connections to the Dedekind zeta function and the analytic class number formula
  • The residue of the Dedekind zeta function at $s=1$ involves the regulator and the class number
• Analogues in function fields and arithmetic geometry, such as the Dirichlet S-unit theorem and the Mordell-Weil theorem for elliptic curves
• Generalizations to the S-unit group and the S-class group in the context of arithmetic geometry and Diophantine geometry

Computational Aspects and Algorithms

• Computing fundamental units and the unit group is a challenging computational problem
• Algorithms based on the geometry of numbers, such as the LLL algorithm and its variants, are used to find a set of fundamental units
• Computation of the regulator involves approximating logarithms of algebraic numbers and computing determinants
• Efficient algorithms for computing the unit group and the class group are essential in computational algebraic number theory
  • Subexponential algorithms, such as the number field sieve and the function field sieve, rely on computing unit groups and class groups
• Connections to the computation of Galois groups, class field theory, and the resolution of Diophantine equations
• Implementations in computer algebra systems, such as PARI/GP, SageMath, and Magma, provide practical tools for working with unit groups and related objects