cyclotomic field

English

Noun

cyclotomic field (plural cyclotomic fields)

${\\displaystyle }$ $\text{[math]}$ (number theory, 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 ${\\displaystyle \\Phi _{n}(x)=\\prod _{\\stackrel {1\\leq k\\leq n}{\\gcd(k,n)=1}}\\left(x-e^{\\frac {2k\\pi i}{n}}\\right)}$ and consequently is a Galois extension of the field of rational numbers.
- 1991, A. Fröhlich, M. J. Taylor, Algebraic Number Theory, Cambridge University Press, 1993, Paperback, page 205,
  Cyclotomic fields are fields obtained by adjoining to ${\\displaystyle \\mathbb {Q} }$ roots of unity, i.e. roots of polynomials of the form ${\\displaystyle X^{n}-1}$ , 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).
- 2007, Israel Kleiner, A History of Abstract Algebra, Springer (Birkhäuser), page 26,
  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.
- 2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 94,
  As in the case of the Drinfel'd double,¹³¹ it can be shown that these fields are subfields of the cyclotomic field ${\\displaystyle \\mathbb {Q} _{N}}$ . Since the Galois group of ${\\displaystyle \\mathbb {Q} _{N}}$ is abelian, every subfield of the cyclotomic field is normal, and consequently preserved by the action of the Galois group of ${\\displaystyle \\mathbb {Q} _{N}}$ .