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| Cyclotomic polynomials | |
|---|---|
| Name | Cyclotomic polynomials |
| Domain | Number theory, Algebra |
| Introduced | 19th century |
| Key people | Carl Friedrich Gauss, Évariste Galois, Ernst Kummer, Richard Dedekind, David Hilbert |
Cyclotomic polynomials are special polynomials that encode the primitive roots of unity and play central roles in Carl Friedrich Gauss's work on constructible polygons, in Évariste Galois's development of group theory, and in modern Richard Dedekind-style algebraic number theory. They arise in the factorization of x^n − 1 and connect to classical results of Ernst Kummer, David Hilbert, Niels Henrik Abel, and contemporary research in Andrew Wiles-adjacent areas. Their structure interacts with objects studied by Srinivasa Ramanujan, John von Neumann, Alexander Grothendieck, and many others across Cambridge University, Princeton University, École Normale Supérieure, and major institutes.
A cyclotomic polynomial Φ_n(x) is defined as the minimal polynomial over the rationals for a primitive n-th root of unity; its degree equals Euler's totient function φ(n), a function studied by Leonhard Euler and later by Pierre-Simon Laplace. For each positive integer n, Φ_n(x) divides x^n − 1 and satisfies the product formula ∏_{d|n} Φ_d(x) = x^n − 1, a relation appearing in the work of Augustin-Louis Cauchy and Joseph-Louis Lagrange. The coefficients of Φ_n(x) are integers, a consequence of Gauss's lemma and properties codified by Richard Dedekind and Leopold Kronecker. Basic symmetry and reciprocity properties relate Φ_n(x) to Φ_n(x^{-1}) and to cyclotomic units studied by Heinrich Weber.
Explicit expressions for Φ_n(x) include formulae using Möbius inversion connected to August Ferdinand Möbius: Φ_n(x) = ∏_{d|n} (x^{n/d} − 1)^{μ(d)}, where μ denotes the Möbius function investigated by Johann Peter Gustav Lejeune Dirichlet. Examples: Φ_1(x)=x−1, Φ_2(x)=x+1, Φ_3(x)=x^2+x+1, and Φ_5(x)=x^4+x^3+x^2+x+1, expressions appearing in classical accounts by Niels Henrik Abel and in constructive polygon proofs by Carl Friedrich Gauss. For prime p the cyclotomic polynomial equals 1 + x + ... + x^{p−1}, a fact used by Galois in early group-theoretic examples and exploited by Évariste Galois's successors in explicit computations at École Polytechnique.
Φ_n(x) is irreducible over the rationals by a theorem attributable to Évariste Galois and later proofs by Ernst Kummer and Richard Dedekind, linking to the study of abelian extensions in Kronecker's Jugendtraum and in Hilbert's class field program. The splitting field of Φ_n(x) yields the n-th cyclotomic field, whose Galois group is isomorphic to the multiplicative group (Z/nZ)^×, a structure analyzed by Leopold Kronecker and David Hilbert. Values of Φ_n at integer arguments relate to multiplicative functions in the work of Srinivasa Ramanujan and Paul Erdős, and discriminant formulas connect to investigations by Heinrich Weber and Emil Artin.
Over the integers Φ_n(x) is irreducible; proofs utilize group actions as in Évariste Galois's framework or reduction modulo primes with methods refined by Ernst Kummer and Helmut Hasse. Over finite fields, Φ_n(x) factors into irreducible polynomials whose degrees divide ord_n(p), a phenomenon exploited in explicit constructions by Goro Shimura and André Weil and used in algorithms developed at Bell Labs and research groups at Massachusetts Institute of Technology. Factors correspond to orbits under the Galois group, echoing themes from Emmy Noether's investigations into invariant theory.
The integer coefficients of Φ_n(x) are bounded but can grow; researchers such as A. Schinzel, Peter Sarnak, Henryk Iwaniec, Elliott H. Lieb, and Ken Ono have contributed estimates on maximal coefficients and height. Lehmer's problem, formulated by D. H. Lehmer and inspiring work by Enrico Bombieri and Harald Helfgott, asks about lower bounds for the Mahler measure of noncyclotomic integer polynomials and intersects cyclotomic study through extreme coefficient behavior. Explicit coefficient computations have been carried out by teams at University of Cambridge and Princeton University and documented in compilations associated with The Royal Society.
Cyclotomic polynomials generate abelian extensions of the rationals and thus underpin class field theory developed by David Hilbert, Emil Artin, and Teiji Takagi, and later expounded by John Tate. The structure of the unit group in cyclotomic fields, involving cyclotomic units, was systematized by Karl Friedrich Gauss successors and studied by Kummer in relation to the first case of Fermat's Last Theorem, a problem historically linked to Andrew Wiles's proof. Relations between Φ_n and reciprocity laws feature in works by Richard Dedekind, Emil Artin, and Helmut Hasse and in modern formulations by Barry Mazur in the context of Iwasawa theory, advanced by Ken Ribet and Ralph Greenberg.
Cyclotomic polynomials appear in constructions of regular polygons as in Carl Friedrich Gauss's 1796 result, in discrete Fourier analysis central to Claude Shannon-era communications and to algorithms by James Cooley and John Tukey, and in coding theory developed at Bell Labs and Bell Laboratories-adjacent research. Generalizations include Gaussian periods studied by Leopold Kronecker and John von Neumann, cyclotomic units in Iwasawa theory, and multivariable analogues arising in research by Pierre Deligne, Alexander Grothendieck, and Maxim Kontsevich. Contemporary applications touch cryptography work by Whitfield Diffie, Ronald Rivest, Adi Shamir, and Leonard Adleman and computational number theory projects at Simons Foundation-funded groups and university laboratories.
Category:Polynomials