Read More
Date: 22-12-2019
1242
Date: 11-11-2019
688
Date: 26-7-2020
1166
|
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,
(1) |
For example, when and , the cyclotomic field is a quadratic field
(2) |
|||
(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 .
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.
|
|
تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
|
|
|
|
|
أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
|
|
|
|
|
قسم شؤون المعارف ينظم دورة عن آليات عمل الفهارس الفنية للموسوعات والكتب لملاكاته
|
|
|