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Cyclotomic Field


A cyclotomic field Q(zeta) is obtained by adjoining a primitive root of unity zeta, say zeta^n=1, to the rational numbers Q. Since zeta is primitive, zeta^k is also an nth root of unity and Q(zeta) contains all of the nth roots of unity,

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

For example, when n=3 and zeta=(-1+isqrt(3))/2, the cyclotomic field is a quadratic field

Q(zeta)={a_0+a_1zeta+a_2zeta^2}
(2)
={b_0+b_1sqrt(-3)}
(3)
=Q(sqrt(-3)),
(4)

where the coefficients b_i are contained in Q.

The Galois group of a cyclotomic field over the rationals is the multiplicative group of Z_n, the ring of integers (mod n). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance, Q(zeta), where zeta^(23)=1.


See also

Extension Field, Number Field

This entry contributed by Todd Rowland

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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.

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Cyclotomic Field

Cite this as:

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

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