The command CyclotomicField(n) creates the n-th cyclotomic field, got by adjoing an n-th root of unity to the rational field.
sage: CyclotomicField(3) Cyclotomic Field of order 3 and degree 2 sage: CyclotomicField(18) Cyclotomic Field of order 18 and degree 6 sage: z = CyclotomicField(6).gen(); z zeta_6 sage: z**3 -1 sage: (1+z)**3 6*zeta_6 - 3
n) |
integral_basis,
multiplicative_order_table,
zeta,
zeta_order
Further documentation:
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Return a list of elements of this number field that are a basis for the full ring of integers.
Instances of class NumberField_cyclotomic also have the following special methods:
__call__,
__repr__,
_latex_
Further documentation:
x) |
The following example illustrates coercion from the cyclotomic field Q(zeta_42) to the cyclotomic field Q(zeta_6), in a case where such coercion is defined:
sage: k42 = CyclotomicField(42) sage: k6 = CyclotomicField(6) sage: a = k42.gen(0) sage: b = a**7 sage: b zeta_42^7 sage: k6(b) zeta_6 sage: b**2 zeta_42^7 - 1 sage: k6(b**2) zeta_6 - 1
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