P-th cyclotomic field
Introduction
A p-th cyclotomic field is a specific type of number field obtained by adjoining a primitive p-th root of unity to the field of rational numbers, denoted as \(\mathbb{Q}\). These fields are of significant interest in algebraic number theory due to their rich structure and applications in various branches of mathematics, including Galois theory, class field theory, and modular forms. The study of cyclotomic fields has been pivotal in the development of modern number theory, particularly in understanding the distribution of prime numbers and the Fermat's Last Theorem.
Definition and Basic Properties
A p-th cyclotomic field is denoted as \(\mathbb{Q}(\zeta_p)\), where \(\zeta_p\) is a primitive p-th root of unity. This means \(\zeta_p\) is a complex number satisfying the equation \(\zeta_p^p = 1\) and \(\zeta_p^k \neq 1\) for \(1 \leq k < p\). The minimal polynomial of \(\zeta_p\) over \(\mathbb{Q}\) is the p-th cyclotomic polynomial, given by:
\[ \Phi_p(x) = \prod_{\substack{1 \leq k < p \\ \gcd(k, p) = 1}} (x - \zeta_p^k) \]
This polynomial is irreducible over \(\mathbb{Q}\) and has degree \(\varphi(p)\), where \(\varphi\) is the Euler's totient function. The field \(\mathbb{Q}(\zeta_p)\) is thus a Galois extension of \(\mathbb{Q}\) with Galois group isomorphic to the multiplicative group of integers modulo p, \((\mathbb{Z}/p\mathbb{Z})^*\).
Algebraic Structure
The ring of integers of \(\mathbb{Q}(\zeta_p)\) is \(\mathbb{Z}[\zeta_p]\), which is the set of all integral linear combinations of powers of \(\zeta_p\). This ring is a Dedekind domain, meaning it has unique factorization of ideals into prime ideals. The structure of the unit group of \(\mathbb{Z}[\zeta_p]\) is given by Dirichlet's unit theorem, which states that the unit group is finitely generated and has rank \(\varphi(p) - 1\).
The discriminant of \(\mathbb{Q}(\zeta_p)\) is given by:
\[ \Delta(\mathbb{Q}(\zeta_p)) = \pm p^{p-2} \]
This discriminant plays a crucial role in the arithmetic of cyclotomic fields, influencing the ramification of primes and the behavior of the field under various extensions.
Galois Group and Automorphisms
The Galois group of \(\mathbb{Q}(\zeta_p)/\mathbb{Q}\) is isomorphic to \((\mathbb{Z}/p\mathbb{Z})^*\), the group of units modulo p. Each element of this group corresponds to an automorphism of \(\mathbb{Q}(\zeta_p)\) that sends \(\zeta_p\) to \(\zeta_p^k\) for some \(k\) coprime to p. The structure of this Galois group is abelian, and its properties are deeply connected to the Kronecker-Weber theorem, which states that every abelian extension of \(\mathbb{Q}\) is contained within a cyclotomic field.
Ramification and Ideal Theory
In the context of cyclotomic fields, ramification refers to how prime numbers in \(\mathbb{Q}\) split into prime ideals in \(\mathbb{Q}(\zeta_p)\). A prime \(q\) in \(\mathbb{Q}\) is said to ramify in \(\mathbb{Q}(\zeta_p)\) if it divides the discriminant \(\Delta(\mathbb{Q}(\zeta_p))\). The only prime that ramifies in \(\mathbb{Q}(\zeta_p)\) is p itself, and it does so completely, meaning that the ideal generated by p in \(\mathbb{Z}[\zeta_p]\) is a power of a single prime ideal.
The splitting of other primes is determined by the Legendre symbol and the behavior of the polynomial \(\Phi_p(x)\) modulo q. If q is a prime not equal to p, it splits into \(\varphi(p)\) distinct prime ideals in \(\mathbb{Q}(\zeta_p)\) if and only if q is congruent to 1 modulo p.
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 object of study in algebraic number theory. For \(\mathbb{Q}(\zeta_p)\), the class number is denoted \(h_p\). The Herbrand-Ribet theorem provides insights into the divisibility of \(h_p\) by p, linking it to the Bernoulli numbers and the Kummer's criterion for the regularity of primes.
The unit group of \(\mathbb{Z}[\zeta_p]\) is an important algebraic object, and its structure is described by the Dirichlet's unit theorem. The units can be expressed in terms of the cyclotomic units, which are specific combinations of powers of \(\zeta_p\). These units form a subgroup of finite index in the full unit group.
Applications and Historical Context
Cyclotomic fields have played a pivotal role in the development of number theory. They were first studied by Carl Friedrich Gauss, who used them to prove the constructibility of the regular p-gon. Later, Ernst Eduard Kummer extended the study of cyclotomic fields to tackle Fermat's Last Theorem, introducing the concept of ideal numbers to deal with the failure of unique factorization.
The study of cyclotomic fields also led to the development of Iwasawa theory, which explores the growth of class numbers in infinite towers of number fields. Cyclotomic fields are also crucial in the study of L-functions and modular forms, linking them to deep conjectures such as the Birch and Swinnerton-Dyer conjecture.