Noun
cyclotomic field (plural cyclotomic fields)
{\displaystyle }
(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
Φ
n
(
x
)
=
∏
gcd
(
k
,
n
)
=
1
1
≤
k
≤
n
(
x
−
e
2
k
π
i
n
)
{\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.