def __iter__(self):
r"""
Create an iterator that generates the elements of `\Q/n\Z` without
repetition, organized by increasing denominator.
For a fixed denominator, elements are listed by increasing numerator.
EXAMPLES:
The first 19 elements of `\Q/5\Z`::
sage: import itertools
sage: list(itertools.islice(QQ/(5*ZZ), 19r))
[0, 1, 2, 3, 4, 1/2, 3/2, 5/2, 7/2, 9/2, 1/3, 2/3, 4/3, 5/3,
7/3, 8/3, 10/3, 11/3, 13/3]
"""
if self.n == 0:
for x in QQ:
yield self(x)
else:
d = ZZ(0)
while True:
for a in d.coprime_integers((d * self.n).floor()):
yield self(a / d)
d += 1
def zeta(self, n=2, all=False):
"""
Return one, or a list of all, primitive n-th root of unity in this unit group.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<z> = NumberField(x^2 + 3)
sage: U = UnitGroup(K)
sage: U.zeta(1)
1
sage: U.zeta(2)
-1
sage: U.zeta(2, all=True)
[-1]
sage: U.zeta(3)
-1/2*z - 1/2
sage: U.zeta(3, all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
sage: U.zeta(4)
Traceback (most recent call last):
...
ValueError: n (=4) does not divide order of generator
sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2+1)
sage: U = UnitGroup(K)
sage: U.zeta(4)
-b
sage: U.zeta(4,all=True)
[-b, b]
sage: U.zeta(3)
Traceback (most recent call last):
...
ValueError: n (=3) does not divide order of generator
sage: U.zeta(3,all=True)
[]
"""
N = self.__ntu
K = self.number_field()
n = ZZ(n)
if n <= 0:
raise ValueError("n (=%s) must be positive" % n)
if n == 1:
if all:
return [K(1)]
else:
return K(1)
elif n == 2:
if all:
return [K(-1)]
else:
return K(-1)
if n.divides(N):
z = self.torsion_generator().value()**(N // n)
if all:
return [z**i for i in n.coprime_integers(n)]
else:
return z
else:
if all:
return []
else:
raise ValueError("n (=%s) does not divide order of generator" %
n)
def zeta(self, n=2, all=False):
"""
Return one, or a list of all, primitive n-th root of unity in this unit group.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<z> = NumberField(x^2 + 3)
sage: U = UnitGroup(K)
sage: U.zeta(1)
1
sage: U.zeta(2)
-1
sage: U.zeta(2, all=True)
[-1]
sage: U.zeta(3)
-1/2*z - 1/2
sage: U.zeta(3, all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
sage: U.zeta(4)
Traceback (most recent call last):
...
ValueError: n (=4) does not divide order of generator
sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2+1)
sage: U = UnitGroup(K)
sage: U.zeta(4)
-b
sage: U.zeta(4,all=True)
[-b, b]
sage: U.zeta(3)
Traceback (most recent call last):
...
ValueError: n (=3) does not divide order of generator
sage: U.zeta(3,all=True)
[]
"""
N = self.__ntu
K = self.number_field()
n = ZZ(n)
if n <= 0:
raise ValueError, "n (=%s) must be positive"%n
if n == 1:
if all:
return [K(1)]
else:
return K(1)
elif n == 2:
if all:
return [K(-1)]
else:
return K(-1)
if n.divides(N):
z = self.torsion_generator().value() ** (N//n)
if all:
return [z**i for i in n.coprime_integers(n)]
else:
return z
else:
if all:
return []
else:
raise ValueError, "n (=%s) does not divide order of generator"%n
