Cyclotomic field
Introduction
A cyclotomic field is a specific type of number field obtained by adjoining a complex primitive root of unity to the field of rational numbers. Cyclotomic fields are significant in algebraic number theory and have applications in various areas of mathematics, including Galois theory, class field theory, and the study of L-functions. The study of cyclotomic fields dates back to the work of Carl Friedrich Gauss, who explored the properties of these fields in the context of constructible polygons and the discriminant of algebraic numbers.
Definition and Basic Properties
A cyclotomic field is denoted as \(\mathbb{Q}(\zeta_n)\), where \(\zeta_n\) is a primitive \(n\)-th root of unity, i.e., a complex number satisfying \(\zeta_n^n = 1\) and \(\zeta_n^k \neq 1\) for \(0 < k < n\). The field \(\mathbb{Q}(\zeta_n)\) is an extension of the rational numbers \(\mathbb{Q}\) and is a Galois extension with a Galois group isomorphic to the multiplicative group \((\mathbb{Z}/n\mathbb{Z})^*\), the group of units modulo \(n\).
The degree of the extension \(\mathbb{Q}(\zeta_n)/\mathbb{Q}\) is given by the Euler's totient function \(\phi(n)\), which counts the number of integers up to \(n\) that are coprime to \(n\). The minimal polynomial of \(\zeta_n\) over \(\mathbb{Q}\) is the \(n\)-th cyclotomic polynomial \(\Phi_n(x)\), which is irreducible over \(\mathbb{Q}\).
Galois Group and Automorphisms
The Galois group of the cyclotomic field \(\mathbb{Q}(\zeta_n)\) over \(\mathbb{Q}\) is isomorphic to \((\mathbb{Z}/n\mathbb{Z})^*\). Each element of this group corresponds to an automorphism of the field, which maps \(\zeta_n\) to \(\zeta_n^k\) for some integer \(k\) coprime to \(n\). This mapping is a homomorphism from \((\mathbb{Z}/n\mathbb{Z})^*\) to the group of automorphisms of \(\mathbb{Q}(\zeta_n)\).
The structure of the Galois group provides insight into the field's properties, such as its subfields and the behavior of its prime ideals. The Kronecker-Weber theorem states that every abelian extension of \(\mathbb{Q}\) is contained within a cyclotomic field, highlighting the fundamental role these fields play in abelian extensions.
Prime Decomposition in Cyclotomic Fields
In cyclotomic fields, the decomposition of prime numbers is a rich area of study. A prime \(p\) in \(\mathbb{Q}\) can either remain prime, split completely, or decompose into a product of prime ideals in \(\mathbb{Q}(\zeta_n)\). The behavior of \(p\) is determined by its congruence class modulo \(n\).
The Dedekind criterion and the use of ramification theory help determine the decomposition of primes in cyclotomic fields. A prime \(p\) is said to be ramified in \(\mathbb{Q}(\zeta_n)\) if it divides \(n\). Unramified primes split according to the order of their residue class in \((\mathbb{Z}/n\mathbb{Z})^*\).
Class Number and Units
The class number of a cyclotomic field, which measures the failure of unique factorization in its ring of integers, is a central topic in algebraic number theory. The class number problem for cyclotomic fields has been extensively studied, with significant contributions from Ernst Eduard Kummer and others.
The unit group of a cyclotomic field is another area of interest. The units of \(\mathbb{Q}(\zeta_n)\) are closely related to the values of Dirichlet characters and the cyclotomic units, which form a subgroup of finite index in the full unit group. The Leopoldt's conjecture and the Vandiver's conjecture are two famous unsolved problems related to the units and class numbers of cyclotomic fields.
Applications and Historical Context
Cyclotomic fields have played a crucial role in the development of modern number theory. Gauss's work on the constructibility of regular polygons led to the discovery of cyclotomic fields, as he showed that a regular \(n\)-gon can be constructed with a compass and straightedge if and only if \(n\) is a product of a power of 2 and distinct Fermat primes.
The study of cyclotomic fields also intersects with the Riemann zeta function and the L-functions, where the values of these functions at negative integers are related to the class numbers of cyclotomic fields. The Birch and Swinnerton-Dyer conjecture and the Iwasawa theory further explore these connections.