# Cyclotomic field

From Citizendium, the Citizens' Compendium

In mathematics, a **cyclotomic field** is a field which is an extension generated by roots of unity. If ζ denotes an *n*-th root of unity, then the *n*-th cyclotomic field *F* is the field extension .

## Contents

## Ring of integers

As above, we take ζ to denote an *n*-th root of unity. The maximal order of *F* is

### Unit group

### Class group

## Splitting of primes

A prime *p* ramifies iff *p* divides *n*. Otherwise, the splitting of *p* depends on the factorisation of the polynomial modulo *p*, which in turn depends on the highest common factor of *p*-1 and *n*.

## Galois group

The minimal polynomial for ζ is the *n*-th cyclotomic polynomial , which is a factor of . Since the powers of ζ are the roots of the latter polynomial, *F* is a splitting field for and hence a Galois extension. The Galois group is isomorphic to the multiplicative group, via

## References

- A. Fröhlich; M.J. Taylor (1991).
*Algebraic number theory*. Cambridge University Press. ISBN 0-521-36664-X. - Serge Lang (1990).
*Cyclotomic Fields I and II*, Combined 2nd edition. Springer-Verlag. ISBN 0-387-96671-4. - Pierre Samuel (1972).
*Algebraic number theory*. Hermann/Kershaw. - I.N. Stewart; D.O. Tall (1979).
*Algebraic number theory*. Chapman and Hall. ISBN 0-412-13840-9. - Lawrence C. Washington (1982).
*Introduction to Cyclotomic Fields*. Springer-Verlag. ISBN 0-387-90622-3.