Meaning of cyclotomic field | Babel Free

Definitions

A number field obtained by adjoining a primitive root of unity to the field of rational numbers.

Equivalents

Examples

“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”

“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).”

“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.”

“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.”

CEFR level