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تسجيل

Cyclotomic Field

المؤلف:  Fröhlich, A. and Taylor, M.

المصدر:  Ch. 6 in Algebraic Number Theory. New York: Cambridge University Press, 1991.

الجزء والصفحة:  ...

16-10-2019

1302

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,

{sum_(k=0)^(n-1)a_izeta^k:a_i in Q}. " src="http://mathworld.wolfram.com/images/equations/CyclotomicField/NumberedEquation1.gif" style="height:47px; width:158px" />

(1)

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

		{a_0+a_1zeta+a_2zeta^2}" src="http://mathworld.wolfram.com/images/equations/CyclotomicField/Inline14.gif" style="height:21px; width:105px" />	(2)
		{b_0+b_1sqrt(-3)}" src="http://mathworld.wolfram.com/images/equations/CyclotomicField/Inline17.gif" style="height:22px; width:87px" />	(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 .

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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.