nLab cyclotomic field

Idea

A field extension $\mathbb{F}(\zeta_n)$ by a primitive root of unity $\zeta_n$.

The default meaning where $\mathbb{F}$ is left unspecified is $\mathbb{F} = \mathbb{Q}$, the field of rational numbers, where $\zeta_n$ may be taken to be $\exp(2\pi i/n)$ (hence the smallest subfield of the complex numbers $\mathbb{C}$ that contains this exponential value).

Related concepts

• cyclotomic integer

• field extension, Galois group

• algebraic numbers

• anti-cyclotomic field

• cyclotomic spectrum

References

• Keith Conrad, Cyclotomic extensions [pdf, pdf]

• Wikipedia, cyclotomic field