def test_AlgebraicNumber_to_root():
assert AlgebraicNumber(sqrt(2)).to_root() == sqrt(2)
zeta5_squared = AlgebraicNumber(CRootOf(x**5 - 1, 4), coeffs=[1, 0, 0])
assert zeta5_squared.to_root() == CRootOf(x**4 + x**3 + x**2 + x + 1, 1)
zeta3_squared = AlgebraicNumber(CRootOf(x**3 - 1, 2), coeffs=[1, 0, 0])
assert zeta3_squared.to_root() == -S(1)/2 - sqrt(3)*I/2
assert zeta3_squared.to_root(radicals=False) == CRootOf(x**2 + x + 1, 0)
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [S.One, S.Zero]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), ()).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), (0, 0)).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(8, 3)]).rep == DMP([QQ(8, 3)], QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(
sqrt(2), [Rational(7, 9), Rational(3, 2)]).rep == DMP([QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [S.One, S(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber( sqrt(3)).rep == DMP([ QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([ QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x, domain='QQ')
assert b.as_poly() == Poly(y, domain='QQ')
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2*p.gen + 3)
assert a.as_poly(x) == Poly(2*x + 3, domain='QQ')
assert b.as_poly() == Poly(2*y + 3, domain='QQ')
assert a.as_expr() == 2*sqrt(2) + 3
assert a.as_expr(x) == 2*x + 3
assert b.as_expr() == 2*sqrt(2) + 3
assert b.as_expr(x) == 2*x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple(1, 0))
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(1, 0), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))
a = AlgebraicNumber(sqrt(2), [1, 2], "alpha")
b = AlgebraicNumber(a)
c = AlgebraicNumber(a, alias="gamma")
assert a == b
assert c.alias.name == "gamma"
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S(1)/2, 0, S(-9)/2, 0])
b = AlgebraicNumber(a, [1, 0, 0])
assert b.root == a.root
assert a.to_root() == sqrt(2)
assert b.to_root() == 2
a = AlgebraicNumber(2)
assert a.is_primitive_element is True