Cyclotomic Field -- from Wolfram MathWorld

Number Theory > Algebraic Number Theory >

MathWorld Contributors > Rowland, Todd >

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,

$Q(zeta)={sum_(k=0)^(n-1)a_izeta^k:a_i in Q}.$

(1)

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

		${a_0+a_1zeta+a_2zeta^2}$	(2)
		${b_0+b_1sqrt(-3)}$	(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 .

This entry contributed by Todd Rowland

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.

CITE THIS AS:

Rowland, Todd. "Cyclotomic Field." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/CyclotomicField.html