def test_str():
# Without alias:
k = QQ.alg_field_from_poly(Poly(x**2 + 7))
frp = k.primes_above(2)[0]
assert str(frp) == '(2, 3*_x/2 + 1/2)'
frp = k.primes_above(3)[0]
assert str(frp) == '(3)'
# With alias:
k = QQ.alg_field_from_poly(Poly(x**2 + 7), alias='alpha')
frp = k.primes_above(2)[0]
assert str(frp) == '(2, 3*alpha/2 + 1/2)'
frp = k.primes_above(3)[0]
assert str(frp) == '(3)'
def test_decomp_7():
# Try working through an AlgebraicField
T = Poly(x**3 + x**2 - 2 * x + 8)
K = QQ.alg_field_from_poly(T)
p = 2
P = K.primes_above(p)
ZK = K.maximal_order()
assert len(P) == 3
assert all(Pi.e == Pi.f == 1 for Pi in P)
assert prod(Pi**Pi.e for Pi in P) == p * ZK
def test_PrimeIdeal_reduce():
k = QQ.alg_field_from_poly(Poly(x**3 + x**2 - 2 * x + 8))
Zk = k.maximal_order()
P = k.primes_above(2)
frp = P[2]
# reduce_element
a = Zk.parent(to_col([23, 20, 11]), denom=6)
a_bar_expected = Zk.parent(to_col([11, 5, 2]), denom=6)
a_bar = frp.reduce_element(a)
assert a_bar == a_bar_expected
# reduce_ANP
a = k([QQ(11, 6), QQ(20, 6), QQ(23, 6)])
a_bar_expected = k([QQ(2, 6), QQ(5, 6), QQ(11, 6)])
a_bar = frp.reduce_ANP(a)
assert a_bar == a_bar_expected
# reduce_alg_num
a = k.to_alg_num(a)
a_bar_expected = k.to_alg_num(a_bar_expected)
a_bar = frp.reduce_alg_num(a)
assert a_bar == a_bar_expected
def test_issue_23402():
k = QQ.alg_field_from_poly(Poly(x**3 + x**2 - 2 * x + 8))
P = k.primes_above(3)
assert P[0].alpha.equiv(0)
