Cyclotomy

Cyclotomy is a branch of number theory that deals with the division of a circle into equal parts and the properties and applications of the roots of unity that arise from this division. The term originates from the Greek words kyklos (circle) and temnein (to cut). Cyclotomy has significant implications in various fields such as cryptography, polynomial theory, and algebraic number theory.

History

The study of cyclotomy can be traced back to ancient mathematicians such as Euclid, who discussed the construction of regular polygons in his work Elements. The problem of cyclotomy was further developed by mathematicians during the Renaissance, including Carl Friedrich Gauss, who made substantial contributions to the theory of cyclotomic fields and cyclotomic polynomials in his work Disquisitiones Arithmeticae.

Cyclotomic Polynomials

A central object of study in cyclotomy is the cyclotomic polynomial, defined as the minimal polynomial over the field of rational numbers (Q) that has a primitive n-th root of unity as a root. The n-th cyclotomic polynomial, denoted as Φ_n(x), is given by: \[ \Phi_n(x) = \prod_{\substack{1 \leq k \leq n \\ \gcd(k, n) = 1}} (x - \zeta_n^k) \] where ζ_n is a primitive n-th root of unity and gcd denotes the greatest common divisor.

Applications

1. 1. 1. Cryptography

In the field of cryptography, cyclotomic polynomials are used in constructing large prime numbers for public key algorithms such as RSA. The properties of roots of unity are also utilized in cryptographic protocols to ensure security and efficiency.

1. 1. 1. Algebraic Number Theory

Cyclotomy is fundamental in the study of algebraic number fields and Galois theory. Cyclotomic fields, which are extension fields generated by a primitive root of unity, are key in understanding the structure of more complex number fields.

1. 1. 1. Polynomial Theory

The factorization properties of cyclotomic polynomials are important in polynomial theory, particularly in the reduction of polynomials over finite fields and the study of polynomial equations.

See Also

• Root of unity
• Euler's totient function
• Galois group
• Field extension