Cyclotomic Field
المؤلف:
Fröhlich, A. and Taylor, M.
المصدر:
Ch. 6 in Algebraic Number Theory. New York: Cambridge University Press, 1991.
الجزء والصفحة:
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16-10-2019
985
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)=<span style=]() {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
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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