def testEasyCase(self):
# Q_rt_minus_1 [1, 0, 1]
field = algfield.NumberField(self.Q_rt_minus_1)
# p=2
Q_rt_minus_1_2_index = [(2, 1)] # ramify
Q_rt_minus_1_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_2_index)
self.assertEqual(result_mul, Q_rt_minus_1_2_prime)
# p=3
Q_rt_minus_1_3_index = [(1, 2)] # inert
Q_rt_minus_1_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_3_index)
self.assertEqual(result_mul, Q_rt_minus_1_3_prime)
# p=5
Q_rt_minus_1_5_index = [(1, 1), (1, 1)] # split
Q_rt_minus_1_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.Q_rt_minus_1))
self.assertEqual(result_e_f, Q_rt_minus_1_5_index)
self.assertEqual(result_mul, Q_rt_minus_1_5_prime)
# Q_cb_2 [-2, 0, 0, 1]
field = algfield.NumberField(self.Q_cb_2)
# p=3
Q_cb_2_3_index = [(3, 1)] # ramify
Q_cb_2_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_3_index)
self.assertEqual(result_mul, Q_cb_2_3_prime)
# p=5
Q_cb_2_5_index = [(1, 1), (1, 2)] # split & inert
Q_cb_2_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_5_index)
self.assertEqual(result_mul, Q_cb_2_5_prime)
# p=31
Q_cb_2_31_index = [(1, 1), (1, 1), (1, 1)] # split
Q_cb_2_31_prime = 31 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(31, self.Q_cb_2))
self.assertEqual(result_e_f, Q_cb_2_31_index)
self.assertEqual(result_mul, Q_cb_2_31_prime)
def _main_prime_decomp(p, polynomial):
"""
main step of prime decomposition
"""
field = algfield.NumberField(polynomial)
n = field.degree
int_basis = field.integer_ring()
base_multiply = module._base_multiplication(int_basis, field)
H_lst = _squarefree_decomp(p, field, base_multiply) # step 2-5
done_list = []
for j in range(len(H_lst)):
H_j = H_lst[j]
# step 7
if H_j != None and H_j.column == n: #rank(H_j)
continue
L = [H_j]
while L:
H = L.pop()
A, gamma_multiply = _separable_algebra(
p, H, base_multiply) # step 8, 9
sol = _check_H(p, H, field, gamma_multiply) # step 10-11
if isinstance(sol, tuple):
done_list.append((sol[0], j + 1, sol[1]))
else:
L += _splitting_squarefree(p, H, base_multiply,
A, gamma_multiply, sol) # step 12-15
return done_list
def testMainCase(self):
# cubic_first [1, 9, 0, 3]
field = algfield.NumberField(self.cubic_first)
# p=3
cubic_first_3_index = [(1, 1), (2, 1)] # split & ramify
cubic_first_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.cubic_first))
self.assertEqual(result_e_f, cubic_first_3_index)
self.assertEqual(result_mul, cubic_first_3_prime)
# cubic_second [1, 8, 0, 3]
field = algfield.NumberField(self.cubic_second)
# p=5
cubic_second_5_index = [(1, 1), (1, 2)] # split & inert
cubic_second_5_prime = 5 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(5, self.cubic_second))
self.assertEqual(result_e_f, cubic_second_5_index)
self.assertEqual(result_mul, cubic_second_5_prime)
# sextic_first [7, 6, 5, 4, 3, 2, 1]
field = algfield.NumberField(self.sextic_first)
# p=2
sextic_first_2_index = [(2, 1), (4, 1)] # split & ramify
sextic_first_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.sextic_first))
self.assertEqual(result_e_f, sextic_first_2_index)
self.assertEqual(result_mul, sextic_first_2_prime)
# sextic_second [6480, 1296, -252, 36, -7, 1]
field = algfield.NumberField(self.sextic_second)
# p=2
sextic_second_2_index = [(1, 1), (1, 4)] # split & ramify
sextic_second_2_prime = 2 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(2, self.sextic_second))
self.assertEqual(result_e_f, sextic_second_2_index)
self.assertEqual(result_mul, sextic_second_2_prime)
# p=3
sextic_second_3_index = [(1, 1), (1, 1), (1, 1), (1, 2)] # split & inert
sextic_second_3_prime = 3 * module.Ideal(
matrix.unitMatrix(field.degree), field, field.integer_ring())
result_e_f, result_mul = self.result_transform(prime_decomp(3, self.sextic_second))
self.assertEqual(result_e_f, sextic_second_3_index)
self.assertEqual(result_mul, sextic_second_3_prime)
def setUp(self):
self.K = algfield.NumberField([-2,0,1])
self.KI = algfield.NumberField([2,0,1])
self.F = algfield.NumberField([-3,0,1])
self.G = algfield.NumberField([-2,0,0,1])
self.CF1 = algfield.NumberField([101,20,1])#cyclotomic field
self.CF2 = algfield.NumberField([14521,-5302,725,-44,1])
def prime_decomp(p, polynomial):
"""
Return prime decomposition of (p) over Q[x]/(polynomial).
This method returns a list of (P_k, e_k, f_k),
where P_k is an instance of Ideal_with_generator,
e_k is the ramification index of P_k,
f_k is the residue degree of P_k.
"""
field = algfield.NumberField(polynomial)
field_disc = field.field_discriminant()
if (field.disc() // field_disc) % p != 0:
return _easy_prime_decomp(p, polynomial)
else:
return _main_prime_decomp(p, polynomial)