def test_minimal_polynomial_rootof(method):
e = RootOf(
x**4 - 3 * x**3 + x**2 * (-3 * sqrt(2) + 1) + 2 * sqrt(2) * x + 2, 0)
assert (minimal_polynomial(e, method=method)(x) == x**8 - 6 * x**7 +
11 * x**6 - 6 * x**5 - 13 * x**4 + 12 * x**3 - 4 * x**2 + 4)
assert minimal_polynomial(e, method=method,
domain=e.poly.domain)(y) == e.poly(y)
def test_abs_re_im(method):
# issue diofant/diofant#662
e1 = abs(sqrt(1 + sqrt(2 + I)))
e2 = re(sqrt(I), evaluate=False)
e3 = im(sqrt(I), evaluate=False)
assert (minimal_polynomial(e1, method=method)(x) == x**16 - 4 * x**12 -
12 * x**8 - 8 * x**4 + 4)
assert minimal_polynomial(e2, method=method)(x) == 2 * x**2 - 1
assert minimal_polynomial(e3, method=method)(x) == 2 * x**2 - 1
def test_diofantissue_662():
e1 = abs(sqrt(1 + sqrt(2 + I)))
e2 = re(sqrt(I), evaluate=False)
e3 = im(sqrt(I), evaluate=False)
for meth in ('compose', 'groebner'):
assert (minimal_polynomial(e1, method=meth)(x) ==
x**16 - 4*x**12 - 12*x**8 - 8*x**4 + 4)
assert minimal_polynomial(e2, method=meth)(x) == 2*x**2 - 1
assert minimal_polynomial(e3, method=meth)(x) == 2*x**2 - 1
def test_minimal_polynomial_sq():
p = expand_multinomial((1 + 5 * sqrt(2) + 2 * sqrt(3))**3)
mp = minimal_polynomial(cbrt(p))(x)
assert mp == x**4 - 4 * x**3 - 118 * x**2 + 244 * x + 1321
p = expand_multinomial((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(cbrt(p))(x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = Add(*[sqrt(i) for i in range(1, 12)])
mp = minimal_polynomial(p)(x)
assert mp.subs({x: 0}) == -71965773323122507776
def test_minimal_polynomial_conjugate():
e = sqrt(1 + sqrt(2 + I))
ec = conjugate(e)
for meth in ('compose', 'groebner'):
assert (minimal_polynomial(ec, method=meth)(x) == minimal_polynomial(
e, method=meth)(x) == x**8 - 4 * x**6 + 2 * x**4 + 4 * x**2 + 2)
assert (minimal_polynomial(
(e + ec) / 2, method=meth)(x) == 4096 * x**16 - 16384 * x**14 +
20480 * x**12 - 12288 * x**10 - 1152 * x**8 + 3328 * x**6 -
1600 * x**4 + 64 * x**2 + 1)
def test_minimal_polynomial_hi_prec():
p = 1 / sqrt(1 - 9 * sqrt(2) + 7 * sqrt(3) + Rational(1, 10)**30)
mp = minimal_polynomial(p)(x)
# checked with Wolfram Alpha
assert mp.coeff(
x**6
) == -1232000000000000000000000000001223999999999999999999999999999987999999999999999999999999999996000000000000000000000000000000
def test_sympyissue_5934():
e = ((-240 * sqrt(2) * sqrt(sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40) -
360 * sqrt(2) * sqrt(-8 * sqrt(5) + 40) * sqrt(-sqrt(5) + 5) -
120 * sqrt(10) * sqrt(-8 * sqrt(5) + 40) * sqrt(-sqrt(5) + 5) +
120 * sqrt(2) * sqrt(-8 * sqrt(5) + 40) * sqrt(sqrt(5) + 5) +
120 * sqrt(2) * sqrt(-sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40) +
120 * sqrt(10) * sqrt(-8 * sqrt(5) + 40) * sqrt(sqrt(5) + 5) +
120 * sqrt(10) * sqrt(-sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40)) /
(-36000 - 7200 * sqrt(5) + (12 * sqrt(10) * sqrt(sqrt(5) + 5) +
24 * sqrt(10) * sqrt(-sqrt(5) + 5))**2))
assert [minimal_polynomial(i)(x) for i in e.as_numer_denom()] == [x] * 2
def test_minpoly_domain():
F = QQ.algebraic_field(sqrt(2))
assert minimal_polynomial(sqrt(2), domain=F) == PurePoly(x - sqrt(2), x, domain=F)
assert minimal_polynomial(sqrt(8), domain=F)(x) == x - 2*sqrt(2)
assert minimal_polynomial(sqrt(Rational(3, 2)), domain=F)(x) == 2*x**2 - 3
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(y, domain=QQ))
# issue sympy/sympy#14494
F = QQ.algebraic_field(I)
assert minimal_polynomial(I, domain=F)(x) == x - I
F = QQ.algebraic_field(sqrt(3)*I)
assert minimal_polynomial(exp(I*pi/3), domain=F)(x) == 2*x - sqrt(3)*I - 1
def test_sympyissue_19760():
e = 1 / (sqrt(1 + sqrt(2)) - sqrt(2) * sqrt(1 + sqrt(2))) + 1
for meth in ('compose', 'groebner'):
minimal_polynomial(e, method=meth)(x) == x**4 - 4 * x**3 + 4 * x**2 - 2
def test_diofantissue_224():
e = (root(root(2, 3) - 1, 3) - root(Rational(1, 9), 3) +
root(Rational(2, 9), 3) - root(Rational(4, 9), 3))
a, b, c, d = e.args
assert minimal_polynomial(Add(d, a, b, c, evaluate=False))(x) == x
assert minimal_polynomial(e)(x) == x
def test_sympyissue_14831():
assert minimal_polynomial(-3 * sqrt(12 * sqrt(2) + 17) + 12 * sqrt(2) +
17 -
2 * sqrt(2) * sqrt(12 * sqrt(2) + 17))(x) == x
def test_minpoly_fraction_field_slow():
assert minimal_polynomial(
minimal_polynomial(sqrt(root(x, 5) - 1))(y).subs(
{y: sqrt(root(x, 5) - 1)}))(z) == z
def test_minpoly_fraction_field():
assert minimal_polynomial(1 / x)(y) == x * y - 1
assert minimal_polynomial(1 / (x + 1))(y) == x * y + y - 1
assert minimal_polynomial(sqrt(x))(y) == y**2 - x
assert minimal_polynomial(sqrt(x), method='groebner')(y) == y**2 - x
assert minimal_polynomial(sqrt(x + 1))(y) == y**2 - x - 1
assert minimal_polynomial(sqrt(x) / x)(y) == x * y**2 - 1
assert minimal_polynomial(sqrt(2) * sqrt(x))(y) == y**2 - 2 * x
assert minimal_polynomial(sqrt(2) + sqrt(x))(y) == \
y**4 - 2*x*y**2 - 4*y**2 + x**2 - 4*x + 4
assert minimal_polynomial(sqrt(2) + sqrt(x), method='groebner')(y) == \
y**4 - 2*x*y**2 - 4*y**2 + x**2 - 4*x + 4
assert minimal_polynomial(cbrt(x))(y) == y**3 - x
assert minimal_polynomial(cbrt(x) + sqrt(x))(y) == \
y**6 - 3*x*y**4 - 2*x*y**3 + 3*x**2*y**2 - 6*x**2*y - x**3 + x**2
assert minimal_polynomial(sqrt(x) / z)(y) == z**2 * y**2 - x
assert minimal_polynomial(
sqrt(x) / (z + 1))(y) == z**2 * y**2 + 2 * z * y**2 + y**2 - x
assert minimal_polynomial(1 / x) == PurePoly(x * y - 1, y)
assert minimal_polynomial(1 / (x + 1)) == PurePoly((x + 1) * y - 1, y)
assert minimal_polynomial(sqrt(x)) == PurePoly(y**2 - x, y)
assert minimal_polynomial(sqrt(x) / z) == PurePoly(z**2 * y**2 - x, y)
# this is (sqrt(1 + x**3)/x).integrate(x).diff(x) - sqrt(1 + x**3)/x
a = sqrt(x) / sqrt(1 + x**(-3)) - sqrt(x**3 + 1) / x + 1 / (
x**Rational(5, 2) *
(1 + x**(-3))**Rational(3, 2)) + 1 / (x**Rational(11, 2) *
(1 + x**(-3))**Rational(3, 2))
assert minimal_polynomial(a)(y) == y
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(exp(x)))
def test_minpoly_sympyissue_7574():
ex = -cbrt(-1) + (-1)**Rational(2, 3)
assert minimal_polynomial(ex)(x) == x + 1
def test_minpoly_sympyissue_7113():
# see discussion in https://github.com/sympy/sympy/pull/2234
r = nsimplify(pi, tolerance=0.000000001)
mp = minimal_polynomial(r)(x)
assert mp == 1768292677839237920489538677417507171630859375*x**109 - \
2734577732179183863586489182929671773182898498218854181690460140337930774573792597743853652058046464
def test_minpoly_compose():
# issue sympy/sympy#6868
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000)) +
2 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000))))
mp = minimal_polynomial(eq + 3)(x)
assert mp == 8000 * x**2 - 48000 * x + 71999
# issue sympy/sympy#5888
assert minimal_polynomial(exp(I * pi / 8))(x) == x**8 + 1
mp = minimal_polynomial(sin(pi / 7) + sqrt(2))(x)
assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
770912*x**4 - 268432*x**2 + 28561
mp = minimal_polynomial(cos(pi / 7) + sqrt(2))(x)
assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
232*x - 239
mp = minimal_polynomial(exp(I * pi / 7) + sqrt(2))(x)
assert mp == x**12 - 2 * x**11 - 9 * x**10 + 16 * x**9 + 43 * x**8 - 70 * x**7 - 97 * x**6 + 126 * x**5 + 211 * x**4 - 212 * x**3 - 37 * x**2 + 142 * x + 127
mp = minimal_polynomial(cos(pi / 9))(x)
assert mp == 8 * x**3 - 6 * x - 1
mp = minimal_polynomial(sin(pi / 7) + sqrt(2))(x)
assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
770912*x**4 - 268432*x**2 + 28561
mp = minimal_polynomial(cos(pi / 7) + sqrt(2))(x)
assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
232*x - 239
mp = minimal_polynomial(exp(I * pi / 7) + sqrt(2))(x)
assert mp == x**12 - 2 * x**11 - 9 * x**10 + 16 * x**9 + 43 * x**8 - 70 * x**7 - 97 * x**6 + 126 * x**5 + 211 * x**4 - 212 * x**3 - 37 * x**2 + 142 * x + 127
mp = minimal_polynomial(exp(2 * I * pi / 7))(x)
assert mp == x**6 + x**5 + x**4 + x**3 + x**2 + x + 1
mp = minimal_polynomial(exp(2 * I * pi / 15))(x)
assert mp == x**8 - x**7 + x**5 - x**4 + x**3 - x + 1
mp = minimal_polynomial(cos(2 * pi / 7))(x)
assert mp == 8 * x**3 + 4 * x**2 - 4 * x - 1
mp = minimal_polynomial(sin(2 * pi / 7))(x)
ex = (5 * cos(2 * pi / 7) - 7) / (9 * cos(pi / 7) - 5 * cos(3 * pi / 7))
mp = minimal_polynomial(ex)(x)
assert mp == x**3 + 2 * x**2 - x - 1
assert minimal_polynomial(-1 /
(2 * cos(pi / 7)))(x) == x**3 + 2 * x**2 - x - 1
assert minimal_polynomial(sin(2*pi/15))(x) == \
256*x**8 - 448*x**6 + 224*x**4 - 32*x**2 + 1
assert minimal_polynomial(sin(5 * pi /
14))(x) == 8 * x**3 - 4 * x**2 - 4 * x + 1
assert minimal_polynomial(cos(
pi / 15))(x) == 16 * x**4 + 8 * x**3 - 16 * x**2 - 8 * x + 1
assert minimal_polynomial(cos(
pi / 17))(x) == (256 * x**8 - 128 * x**7 - 448 * x**6 + 192 * x**5 +
240 * x**4 - 80 * x**3 - 40 * x**2 + 8 * x + 1)
assert minimal_polynomial(cos(
2 * pi / 21))(x) == (64 * x**6 - 32 * x**5 - 96 * x**4 + 48 * x**3 +
32 * x**2 - 16 * x + 1)
ex = RootOf(x**3 + x * 4 + 1, 0)
mp = minimal_polynomial(ex)(x)
assert mp == x**3 + 4 * x + 1
mp = minimal_polynomial(ex + 1)(x)
assert mp == x**3 - 3 * x**2 + 7 * x - 4
assert minimal_polynomial(exp(I * pi / 3))(x) == x**2 - x + 1
assert minimal_polynomial(exp(I * pi / 4))(x) == x**4 + 1
assert minimal_polynomial(exp(I * pi / 6))(x) == x**4 - x**2 + 1
assert minimal_polynomial(exp(I * pi / 9))(x) == x**6 - x**3 + 1
assert minimal_polynomial(exp(I * pi /
10))(x) == x**8 - x**6 + x**4 - x**2 + 1
assert minimal_polynomial(exp(I * pi / 18))(x) == x**12 - x**6 + 1
assert minimal_polynomial(sin(
pi / 9))(x) == 64 * x**6 - 96 * x**4 + 36 * x**2 - 3
assert minimal_polynomial(sin(pi/11))(x) == 1024*x**10 - 2816*x**8 + \
2816*x**6 - 1232*x**4 + 220*x**2 - 11
ex = cbrt(2) * exp(Rational(2, 3) * I * pi)
assert minimal_polynomial(ex)(x) == x**3 - 2
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(cos(pi * sqrt(2))))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(pi * sqrt(2))))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(exp(I * pi * sqrt(2))))
# issue sympy/sympy#5934
ex = 1 / (-36000 - 7200 * sqrt(5) +
(12 * sqrt(10) * sqrt(sqrt(5) + 5) +
24 * sqrt(10) * sqrt(-sqrt(5) + 5))**2) + 1
pytest.raises(ZeroDivisionError, lambda: minimal_polynomial(ex))
ex = sqrt(1 + cbrt(2)) + sqrt(1 + root(2, 4)) + sqrt(2)
mp = minimal_polynomial(ex)(x)
assert degree(mp) == 48 and mp.subs({x: 0}) == -16630256576
mp = minimal_polynomial(sin(pi / 27))(x)
assert mp == (262144 * x**18 - 1179648 * x**16 + 2211840 * x**14 -
2236416 * x**12 + 1317888 * x**10 - 456192 * x**8 +
88704 * x**6 - 8640 * x**4 + 324 * x**2 - 3)
ex = sqrt(2) - RootOf(x**2 - 2, 0, radicals=False)
for meth in ('compose', 'groebner'):
assert minimal_polynomial(ex, method=meth)(x) == x**2 - 8
def test_sympyissue_19760(method):
e = 1 / (sqrt(1 + sqrt(2)) - sqrt(2) * sqrt(1 + sqrt(2))) + 1
assert minimal_polynomial(
e, method=method)(x) == x**4 - 4 * x**3 + 4 * x**2 - 2
def test_minimal_polynomial_GoldenRatio(method):
assert minimal_polynomial(GoldenRatio, method=method)(x) == x**2 - x - 1
def test_sympyissue_22400(method):
e = root(2, 3) + root(3, 3) + (-1 + I * sqrt(3)) / 2 * root(5, 3)
r = (x**27 - 90 * x**24 + 1089 * x**21 - 62130 * x**18 + 105507 * x**15 -
16537410 * x**12 - 30081453 * x**9 - 1886601330 * x**6 +
73062900 * x**3 - 6859000)
assert minimal_polynomial(e, method=method)(x) == r
def test_minimal_polynomial():
assert minimal_polynomial(-7)(x) == x + 7
assert minimal_polynomial(-1)(x) == x + 1
assert minimal_polynomial(+0)(x) == x
assert minimal_polynomial(+1)(x) == x - 1
assert minimal_polynomial(+7)(x) == x - 7
assert minimal_polynomial(Rational(1, 3),
method='groebner')(x) == 3 * x - 1
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(pi, method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(sin(sqrt(2)), method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(2**pi, method='groebner'))
pytest.raises(ValueError, lambda: minimal_polynomial(1, method='spam'))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(5))(x) == x**2 - 5
assert minimal_polynomial(sqrt(6))(x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2))(x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5))(x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6))(x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3)(x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6)(x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7)(x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3)(x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6)(x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7)(x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)))(x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)))(x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I +
sqrt(2 + I))(x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3))(x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) +
sqrt(6))(x) == x**4 - 22 * x**2 - 48 * x - 23
e = 1 / sqrt(sqrt(1 + sqrt(3)) - 4)
assert minimal_polynomial(e) == minimal_polynomial(e, method='groebner')
assert minimal_polynomial(e)(x) == (222 * x**8 + 240 * x**6 + 94 * x**4 +
16 * x**2 + 1)
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a)(x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a))(x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1)))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(2)) == PurePoly(x**2 - 2)
assert minimal_polynomial(sqrt(2), method='groebner') == PurePoly(x**2 - 2)
a = sqrt(2)
b = sqrt(3)
assert minimal_polynomial(b)(x) == x**2 - 3
assert minimal_polynomial(a) == PurePoly(x**2 - 2)
assert minimal_polynomial(b) == PurePoly(x**2 - 3)
assert minimal_polynomial(sqrt(a)) == PurePoly(x**4 - 2)
assert minimal_polynomial(a + 1) == PurePoly(x**2 - 2 * x - 1)
assert minimal_polynomial(sqrt(a / 2 +
17))(x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 +
17))(x) == 4 * x**4 - 136 * x**2 + 1153
# issue diofant/diofant#431
K = QQ.algebraic_field(sqrt(2))
theta = K.to_expr(K([17, Rational(1, 2)]))
assert minimal_polynomial(theta)(x) == 2 * x**2 - 68 * x + 577
K = QQ.algebraic_field(RootOf(x**7 + x - 1, 3))
theta = K.to_expr(K([1, 0, 0, 2, 1]))
ans = minimal_polynomial(theta)(x)
assert ans == (x**7 - 7 * x**6 + 19 * x**5 - 27 * x**4 + 63 * x**3 -
115 * x**2 + 82 * x - 147)
assert minimal_polynomial(theta, method='groebner')(x) == ans
K = QQ.algebraic_field(RootOf(x**5 + 5 * x - 1, 2))
theta = K.to_expr(K([1, -1, 1]))
ans = (x**30 - 15 * x**28 - 10 * x**27 + 135 * x**26 + 330 * x**25 -
705 * x**24 - 150 * x**23 + 3165 * x**22 - 6850 * x**21 +
7182 * x**20 + 3900 * x**19 + 4435 * x**18 + 11970 * x**17 -
259725 * x**16 - 18002 * x**15 + 808215 * x**14 - 200310 * x**13 -
647115 * x**12 + 299280 * x**11 - 1999332 * x**10 + 910120 * x**9 +
2273040 * x**8 - 5560320 * x**7 + 5302000 * x**6 - 2405376 * x**5 +
1016640 * x**4 - 804480 * x**3 + 257280 * x**2 - 53760 * x + 1280)
assert minimal_polynomial(sqrt(theta) + root(theta, 3),
method='groebner')(x) == ans
K1 = QQ.algebraic_field(RootOf(x**3 + 4 * x - 15, 1))
K2 = QQ.algebraic_field(RootOf(x**3 - x + 1, 0))
theta = sqrt(1 + 1 / (K1.to_expr(K1([1, 0, 1])) + 1 /
(sqrt(3) + K2.to_expr(K2([-1, 2, 1])))))
ans = (2262264837876687263 * x**36 - 38939909597855051866 * x**34 +
315720420314462950715 * x**32 - 1601958657418182606114 * x**30 +
5699493671077371036494 * x**28 - 15096777696140985506150 * x**26 +
30847690820556462893974 * x**24 - 49706549068200640994022 * x**22 +
64013601241426223813103 * x**20 - 66358713088213594372990 * x**18 +
55482571280904904971976 * x**16 - 37309340229165533529076 * x**14 +
20016999328983554519040 * x**12 - 8446273798231518826782 * x**10 +
2738866994867366499481 * x**8 - 657825125060873756424 * x**6 +
110036313740049140508 * x**4 - 11416087328869938298 * x**2 +
551322649782053543)
assert minimal_polynomial(theta)(x) == ans
a = sqrt(2) / 3 + 7
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)))(x) == f
assert minimal_polynomial(a**Rational(
3, 2))(x) == 729 * x**4 - 506898 * x**2 + 84604519
K = QQ.algebraic_field(RootOf(x**3 + x - 1, 0))
a = K.to_expr(K([0, 1]))
assert minimal_polynomial(1 / a**2)(x) == x**3 - x**2 - 2 * x - 1
# issue sympy/sympy#5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000)) +
2 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000))))
assert minimal_polynomial(eq)(x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex)(x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex)(x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = cbrt(expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(-sqrt(5) / 2 - Rational(1, 2) +
(-sqrt(5) / 2 - Rational(1, 2))**2)(x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3)(x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I,
method='groebner')(x) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1,
method='groebner')(x) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
method='groebner')(x) == x**4 + 18 * x**2 + 49
assert minimal_polynomial(exp_polar(0))(x) == x - 1