Dictionary

cyclotomic field en · NOUN

Meanings

1. (qualifier:algebraic number theory) A number field obtained by adjoining a primitive root of unity to the field of rational numbers.
   • A cyclotomic field is the splitting field of the cyclotomic polynomial #92;Phi#95;n(x)#61;#92;prod#95;#92;stackrel#123;1#92;lek#92;len#125;#123;#92;gcd(k,n)#61;1#125;#92;left(x-e#123;#92;frac#123;2k#92;pii#125;#123;n
   • At about the same time Kummer introduced his "ideal numbers," defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field—in our terminology, the order of the ideal class group of the cyclotomic field. 2007, Israel Kleiner, A History of Abstract Algebra, Springer (Birkhäuser), page 26:
   • As in the case of the Drinfel'd double,¹³¹ it can be shown that these fields are subfields of the cyclotomic field #92;Q#95;N. Since the Galois group of #92;Q#95;N is abelian, every subfield of the cyclotomic field is normal, and consequently preserved by the action of the Galois group of #92;Q#95;N. 2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 94:
   • Cyclotomic fields are fields obtained by adjoining to #92;Q roots of unity, i.e. roots of polynomials of the form Xⁿ-1, although the reader is warned that this terminology will be extended in §2.[…]Cyclotomic fields play a fundamental role in a number of arithmetic problems: for instance primes in arithmetic progression (see VIII,§4) and Fermat's Last Theorem (VII,§1). 1991, A. Fröhlich, M. J. Taylor, “Algebraic Number Theory”, in Paperback, Cambridge University Press, published 1993, page 205:

Forms

Hypernyms

number field

Meronyms

Kummer ring · cyclotomic integer

Relateds

quadratic field · cyclotomic polynomial

Translations (5)

de zyklotomische Körper (extension field of the rational numbers formed by adjoining a primitive root of unity) · de Kreisteilungskörper (extension field of the rational numbers formed by adjoining a primitive root of unity) · pl ciało cyklotomiczne (extension field of the rational numbers formed by adjoining a primitive root of unity) · it campo ciclotomico (extension field of the rational numbers formed by adjoining a primitive root of unity) · es cuerpo ciclotómico (extension field of the rational numbers formed by adjoining a primitive root of unity)