def test_field_isomorphism():
assert field_isomorphism(3, sqrt(2)) == [3]
assert field_isomorphism( I*sqrt(3), I*sqrt(3)/2) == [ 2, 0]
assert field_isomorphism(-I*sqrt(3), I*sqrt(3)/2) == [-2, 0]
assert field_isomorphism( I*sqrt(3), -I*sqrt(3)/2) == [-2, 0]
assert field_isomorphism(-I*sqrt(3), -I*sqrt(3)/2) == [ 2, 0]
assert field_isomorphism( 2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [ S(6)/35, 0]
assert field_isomorphism(-2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [-S(6)/35, 0]
assert field_isomorphism( 2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [-S(6)/35, 0]
assert field_isomorphism(-2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [ S(6)/35, 0]
assert field_isomorphism(
2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [ S(6)/35, 27]
assert field_isomorphism(
-2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [-S(6)/35, 27]
assert field_isomorphism(
2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [-S(6)/35, 27]
assert field_isomorphism(
-2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [ S(6)/35, 27]
p = AlgebraicNumber( sqrt(2) + sqrt(3))
q = AlgebraicNumber(-sqrt(2) + sqrt(3))
r = AlgebraicNumber( sqrt(2) - sqrt(3))
s = AlgebraicNumber(-sqrt(2) - sqrt(3))
pos_coeffs = [ S(1)/2, S(0), -S(9)/2, S(0)]
neg_coeffs = [-S(1)/2, S(0), S(9)/2, S(0)]
a = AlgebraicNumber(sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = AlgebraicNumber(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [ S(1)/2, S(0), -S(11)/2, S(0)]
neg_coeffs = [-S(1)/2, S(0), S(11)/2, S(0)]
a = AlgebraicNumber(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [ S(3)/2, S(0), -S(33)/2, -S(8)]
neg_coeffs = [-S(3)/2, S(0), S(33)/2, -S(8)]
a = AlgebraicNumber(3*sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(3*sqrt(2) + 2*sqrt(3) + 1)
pos_1_coeffs = [ S(1)/2, S(0), -S(5)/2, S(1)]
neg_5_coeffs = [-S(5)/2, S(0), S(49)/2, S(1)]
pos_5_coeffs = [ S(5)/2, S(0), -S(49)/2, S(1)]
neg_1_coeffs = [-S(1)/2, S(0), S(5)/2, S(1)]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
def test_field_isomorphism():
assert field_isomorphism(3, sqrt(2)) == [3]
assert field_isomorphism(I * sqrt(3), I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(-I * sqrt(3), I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(I * sqrt(3), -I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(-I * sqrt(3), -I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(2 * I * sqrt(3) / 7,
5 * I * sqrt(3) / 3) == [S(6) / 35, 0]
assert field_isomorphism(-2 * I * sqrt(3) / 7,
5 * I * sqrt(3) / 3) == [-S(6) / 35, 0]
assert field_isomorphism(2 * I * sqrt(3) / 7,
-5 * I * sqrt(3) / 3) == [-S(6) / 35, 0]
assert field_isomorphism(-2 * I * sqrt(3) / 7,
-5 * I * sqrt(3) / 3) == [S(6) / 35, 0]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [S(6) / 35, 27]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [-S(6) / 35, 27]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [-S(6) / 35, 27]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [S(6) / 35, 27]
p = AlgebraicNumber(sqrt(2) + sqrt(3))
q = AlgebraicNumber(-sqrt(2) + sqrt(3))
r = AlgebraicNumber(sqrt(2) - sqrt(3))
s = AlgebraicNumber(-sqrt(2) - sqrt(3))
pos_coeffs = [S(1) / 2, S(0), -S(9) / 2, S(0)]
neg_coeffs = [-S(1) / 2, S(0), S(9) / 2, S(0)]
a = AlgebraicNumber(sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = AlgebraicNumber(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [S(1) / 2, S(0), -S(11) / 2, S(0)]
neg_coeffs = [-S(1) / 2, S(0), S(11) / 2, S(0)]
a = AlgebraicNumber(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [S(3) / 2, S(0), -S(33) / 2, -S(8)]
neg_coeffs = [-S(3) / 2, S(0), S(33) / 2, -S(8)]
a = AlgebraicNumber(3 * sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(3 * sqrt(2) + 2 * sqrt(3) + 1)
pos_1_coeffs = [S(1) / 2, S(0), -S(5) / 2, S(1)]
neg_5_coeffs = [-S(5) / 2, S(0), S(49) / 2, S(1)]
pos_5_coeffs = [S(5) / 2, S(0), -S(49) / 2, S(1)]
neg_1_coeffs = [-S(1) / 2, S(0), S(5) / 2, S(1)]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
def test_field_isomorphism():
assert field_isomorphism(3, sqrt(2)) == [3]
assert field_isomorphism(I * sqrt(3), I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(-I * sqrt(3), I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(I * sqrt(3), -I * sqrt(3) / 2) == [-2, 0]
assert field_isomorphism(-I * sqrt(3), -I * sqrt(3) / 2) == [2, 0]
assert field_isomorphism(2 * I * sqrt(3) / 7, 5 * I * sqrt(3) / 3) == [
Rational(6, 35),
0,
]
assert field_isomorphism(-2 * I * sqrt(3) / 7, 5 * I * sqrt(3) / 3) == [
Rational(-6, 35),
0,
]
assert field_isomorphism(2 * I * sqrt(3) / 7, -5 * I * sqrt(3) / 3) == [
Rational(-6, 35),
0,
]
assert field_isomorphism(-2 * I * sqrt(3) / 7, -5 * I * sqrt(3) / 3) == [
Rational(6, 35),
0,
]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [
Rational(6, 35),
27,
]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
5 * I * sqrt(3) / 3) == [
Rational(-6, 35),
27,
]
assert field_isomorphism(2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [
Rational(-6, 35),
27,
]
assert field_isomorphism(-2 * I * sqrt(3) / 7 + 27,
-5 * I * sqrt(3) / 3) == [
Rational(6, 35),
27,
]
p = AlgebraicNumber(sqrt(2) + sqrt(3))
q = AlgebraicNumber(-sqrt(2) + sqrt(3))
r = AlgebraicNumber(sqrt(2) - sqrt(3))
s = AlgebraicNumber(-sqrt(2) - sqrt(3))
pos_coeffs = [S.Half, S.Zero, Rational(-9, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(9, 2), S.Zero]
a = AlgebraicNumber(sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = AlgebraicNumber(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [S.Half, S.Zero, Rational(-11, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(11, 2), S.Zero]
a = AlgebraicNumber(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [Rational(3, 2), S.Zero, Rational(-33, 2), -S(8)]
neg_coeffs = [Rational(-3, 2), S.Zero, Rational(33, 2), -S(8)]
a = AlgebraicNumber(3 * sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(3 * sqrt(2) + 2 * sqrt(3) + 1)
pos_1_coeffs = [S.Half, S.Zero, Rational(-5, 2), S.One]
neg_5_coeffs = [Rational(-5, 2), S.Zero, Rational(49, 2), S.One]
pos_5_coeffs = [Rational(5, 2), S.Zero, Rational(-49, 2), S.One]
neg_1_coeffs = [Rational(-1, 2), S.Zero, Rational(5, 2), S.One]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
