cyclotomische

Cyclotomische refers to concepts and objects in mathematics that are related to cyclotomic polynomials and cyclotomic fields. The adjective is the German form of the English term “cyclotomic,” which itself is derived from the Greek words “kyklos” (circle) and “tomos” (cut), reflecting the division of the circle by roots of unity. In number theory, cyclotomic fields are finite extensions of the rational numbers obtained by adjoining a primitive nth root of unity to ℚ. The ring of integers in a cyclotomic field is called a cyclotomic integer, and such integers have many important algebraic properties.

The cyclotomic polynomial Φn(x) is the monic polynomial whose roots are the primitive nth roots of unity. It satisfies the relation xⁿ – 1 = ∏d|n Φd(x) and has degree φ(n), where φ is Euler’s totient function. Cyclotomische polynomials play a central role in the study of field extensions, Galois theory, and the proof of the Kronecker–Weber theorem, which states that every finite abelian extension of ℚ is contained in a cyclotomic field.

Beyond pure mathematics, cyclotomische concepts appear in coding theory, cryptography, and the construction of complex Hadamard matrices. The ubiquity of cyclotomic structures in algebraic number theory and discrete mathematics highlights their foundational significance.