Cyclotomic Field

A cyclotomic field  is obtained by adjoining a primitive root of unity , say , to the rational numbers . Since  is primitive,  is also an th root of unity and  contains all of the th roots of unity,

(1)

For example, when  and , the cyclotomic field is a quadratic field

			(2)
			(3)
			(4)

where the coefficients  are contained in .

The Galois group of a cyclotomic field over the rationals is the multiplicative group of , the ring of integers (mod ). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance, , where .

REFERENCES:

Fröhlich, A. and Taylor, M. Ch. 6 in Algebraic Number Theory. New York: Cambridge University Press, 1991.

Koch, H. "Cyclotomic Fields." §6.4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 180-184, 2000.

Weiss, E. Algebraic Number Theory. New York: Dover, 1998.