Cyclotomic units
Introduction
Cyclotomic units are a fundamental concept in algebraic number theory, particularly in the study of cyclotomic fields. These units are elements of the group of units in the ring of integers of a cyclotomic field, which are generated by roots of unity. Cyclotomic units play a crucial role in the study of class numbers, Iwasawa theory, and the arithmetic of cyclotomic fields. They provide insight into the structure of the unit group and are instrumental in formulating and proving various conjectures and theorems in number theory.
Cyclotomic Fields
A cyclotomic field is a number field obtained by adjoining a primitive nth root of unity to the field of rational numbers, denoted as \(\mathbb{Q}(\zeta_n)\), where \(\zeta_n\) is a primitive nth root of unity. The ring of integers of a cyclotomic field is the integral closure of \(\mathbb{Z}\) in \(\mathbb{Q}(\zeta_n)\). The study of cyclotomic fields is deeply intertwined with the properties of cyclotomic units.
Definition and Construction
Cyclotomic units are specific units in the ring of integers of a cyclotomic field. They are constructed using the roots of unity and are defined as follows:
For a primitive nth root of unity \(\zeta_n\), the cyclotomic units are generated by elements of the form:
\[ \eta_k = \frac{1 - \zeta_n^k}{1 - \zeta_n} \]
for \(1 \leq k < n\), where \(k\) is coprime to \(n\). These units are part of the larger group of units in the cyclotomic field, and they form a subgroup known as the cyclotomic unit group.
Properties of Cyclotomic Units
Cyclotomic units have several important properties:
1. **Norm and Trace**: The norm and trace of cyclotomic units can be explicitly calculated, providing valuable information about the arithmetic of the field.
2. **Regulators**: The logarithmic embedding of cyclotomic units into the real numbers gives rise to the concept of regulators, which measure the "size" of the unit group in a certain sense.
3. **Density**: Cyclotomic units are dense in the unit group of the cyclotomic field, meaning that they approximate other units closely.
4. **Relation to Class Numbers**: Cyclotomic units are closely related to the class number of the cyclotomic field, which is a measure of the failure of unique factorization.
Applications in Number Theory
Cyclotomic units are used in various areas of number theory:
1. **Iwasawa Theory**: Cyclotomic units are central to Iwasawa theory, which studies the growth of class numbers in infinite extensions of number fields.
2. **Kummer Theory**: In Kummer theory, cyclotomic units help in understanding the splitting of primes in extensions of cyclotomic fields.
3. **L-functions and Zeta Functions**: Cyclotomic units appear in the study of special values of L-functions and zeta functions, particularly in the context of the Birch and Swinnerton-Dyer conjecture.
4. **Galois Cohomology**: Cyclotomic units are used in the study of Galois cohomology, providing insight into the structure of Galois groups of cyclotomic fields.
Computational Aspects
The computation of cyclotomic units involves several steps:
1. **Root of Unity Calculations**: Calculating powers and combinations of roots of unity to construct cyclotomic units.
2. **Unit Group Structure**: Determining the structure of the unit group in the cyclotomic field, often using computational algebra systems.
3. **Regulator Calculations**: Computing the regulator of the cyclotomic unit group, which involves logarithmic embeddings.