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Date: 23-1-2020
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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,
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For example, when and , the cyclotomic field is a quadratic field
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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.
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