Cyclotomic FieldsEdit
Cyclotomic fields are a cornerstone of algebraic number theory. They arise by adjoining a primitive nth root of unity to the rational numbers, and they sit at the crossroads of concrete arithmetic and abstract symmetry. Formally, if ζn denotes a primitive nth root of unity, the cyclotomic field is Q(ζn). This field is the splitting field of the polynomial x^n − 1 over Q and, for a primitive root, has degree φ(n), the Euler totient. The ring of integers in Q(ζn) is Z[ζn], and the subfields generated by various combinations of roots of unity reflect the arithmetic of the integers modulo n. In the language of links, cyclotomic fields are intimately connected to the theory of roots of unity, Dirichlet characters, and abelian extensions of Q Cyclotomic polynomial Root of unity.
These fields are not just curiosities; they encode a great deal of arithmetic information in a particularly explicit way. The Galois group Gal(Q(ζn)/Q) is naturally isomorphic to (Z/nZ)×, with automorphisms acting by ζn ↦ ζn^a for a in (Z/nZ)×. This explicit Galois action makes cyclotomic fields exemplary test cases for broader principles in Galois theory and class field theory. The real subfield Q(ζn + ζn^−1) provides a standard way to study totally real subfields of cyclotomic fields, and the full field itself contains all nth roots of unity. These features connect directly to the analytic side of number theory through Dirichlet L-functions and character sums, since characters modulo n give a concrete way to package arithmetic information into L-series. See for example Dirichlet characters Dirichlet character and Dirichlet L-functions Dirichlet L-function.
Definition and basic properties
- Construction and generators: A primitive nth root of unity ζn satisfies ζn^n = 1 and no smaller positive power equals 1. The field Q(ζn) is the smallest field containing Q and all the nth roots of unity; it is the splitting field of x^n − 1 over Q. The nth cyclotomic polynomial Φn(x) is the minimal polynomial of ζn over Q, and Q(ζn) = Q(ζn) is generated by a single element, ζn. Because Φn has degree φ(n), the field Q(ζn) has degree φ(n) over Q. The disciminant and ramification properties of Q(ζn) are governed by n: primes dividing n ramify, and primes p ∤ n split according to congruence conditions modulo n.
- Ring of integers: The integers of Q(ζn) form the ring Z[ζn]. This presentation makes explicit computations possible in many cases, and it is a standard reference point for algorithmic number theory.
- Cyclotomic polynomials: The polynomials Φn(x) encode the minimal polynomials of primitive nth roots of unity and give a compact way to describe the splitting field. See cyclotomic polynomials Cyclotomic polynomial for related material.
- Subfields and real forms: The real subfield Q(ζn + ζn^−1) has degree φ(n)/2 for n > 2 and provides a natural bridge to totally real arithmetic within the cyclotomic tower. This contrasts with the full field, which is generally a CM-field (a totally imaginary quadratic extension of a totally real subfield).
Galois structure and subfields
- Galois group: Gal(Q(ζn)/Q) ≅ (Z/nZ)×, and the action is by automorphisms given by ζn ↦ ζn^a with a ∈ (Z/nZ)×. This explicit description makes cyclotomic fields one of the most transparent examples of the Galois correspondence in action. The structure of the group reflects arithmetic properties of modulo n, linking to primitive roots and residue classes.
- Subfields and abelian extensions: Every subgroup of the Galois group corresponds to a subfield of Q(ζn), and vice versa. These subfields often have their own arithmetic interest, and they collectively encode the abelian extensions of Q. The Kronecker-Weber theorem identifies the landscape: every finite abelian extension of Q sits inside some cyclotomic field Kronecker–Weber theorem.
- Real vs complex: The full cyclotomic field is a complex field in general, while the real subfield is obtained by taking fixed fields under complex conjugation. This dichotomy underpins many explicit computations and helps separate real arithmetical phenomena from purely imaginary phenomena within the same framework.
Arithmetic and class field theory connections
- Ramification and discriminants: For n > 1, primes p dividing n ramify in Q(ζn), while primes p ∤ n have decomposition behavior dictated by congruence mod n. The discriminant of Q(ζn) is given by a classical formula, Disc(Q(ζn)) = (−1)^{φ(n)/2} n^{φ(n)} / ∏_{p|n} p^{φ(n)/(p−1)}; this encodes how primes ramify in the field.
- Units and arithmetic invariants: The unit group of Z[ζn] and the structure of the ideal class group of Q(ζn) are central to various lines of inquiry, from explicit class field theory to computations of regulators and L-values. In particular, cyclotomic fields provided the testing ground for Kummer’s work on FLT through the study of ideal class groups in these fields.
- Dirichlet characters and L-functions: The arithmetic of Q(ζn) is closely tied to Dirichlet characters modulo n. These characters appear in the explicit description of the Galois action and in the analytic study of L-functions, which encode information about primes in arithmetic progressions and relate to the distribution of primes via the analytic class number formula. See Dirichlet characters Dirichlet character and Dirichlet L-functions Dirichlet L-function for broader context.
- Class field theory and explicit reciprocity: Cyclotomic fields are a central, explicitly computable instance of the ideas in class field theory. The Kronecker-Weber theorem is the classical backbone of this viewpoint, while modern developments such as Iwasawa theory extend the cyclotomic perspective to infinite towers of fields Class field theory Iwasawa theory.
Historical development and perspective
- Origins and Gauss: The study of cyclotomic fields grew out of Gauss’s work on cyclotomic integers and the solvability of equations by radicals. The explicit nature of the roots of unity provided a concrete laboratory for exploring the relationship between arithmetic and symmetry.
- Kummer, Fermat, and regular primes: In the 19th century, Kummer developed deep techniques for studying FLT by examining the arithmetic of cyclotomic fields, notably the pth cyclotomic field Q(ζp). He showed that irregular primes—those p dividing the class number of Q(ζp)—pose obstructions to certain factorizations. This line of inquiry led to important conjectures and partial results, including the notion of regular primes and the Vandiver conjecture Kummer, Fermat's Last Theorem, Vandiver conjecture.
- The Kronecker-Weber milestone and modern theory: The discovery that all finite abelian extensions of Q are contained in cyclotomic fields unified a broad swath of algebraic number theory under a single, explicit construction. This opened the door to modern class field theory, Iwasawa theory, and the analytic study of L-functions, turning cyclotomic fields into a foundational tool for both explicit calculation and deep structural insights Kronecker–Weber theorem.
- Contemporary view and debates: The development of the subject reflects a broader mathematical posture that prizes explicit structure and long-range applicability. Critics of overly abstract machinery sometimes argued for a tighter focus on problems with direct, practical impact; supporters contend that the cyclotomic framework yields durable tools, makes symmetry and arithmetic tangible, and underpins advances in cryptography, algorithmic number theory, and beyond. The interplay between concrete arithmetic of roots of unity and abstract frameworks such as class field theory remains a productive balance point in the discipline Class field theory Iwasawa theory.