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The minimum polynomial of Power[2, (3)^-1] (-(1/2) + (I Sqrt[3])/2) Python minimal_polynomial function can get the result as x^2 + 2^(1/3)*x + 2^(2/3) .(In Python ** means^) Python

import os
import sympy as sy
from sympy import minimal_polynomial, sqrt, solve, QQ, Rational
#Rational

from sympy.abc import x, y, z
s = minimal_polynomial (2 ** Rational (1, 
     3)*(Rational (-1, 2) + 
      Rational (1, 2)*3 ** Rational (1, 2)*sqrt (-1)), x, 
   domain = QQ.algebraic_field (2 ** Rational (1, 3)))
print (s)
print (sy.latex (s))

But MMA can't get the result.

MinimalPolynomial[Power[2, (3)^-1] (-(1/2) + (I Sqrt[3])/2), x, 
 Extension -> Power[2, (3)^-1]]
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    $\begingroup$ This problem may be a bug of MMA. $\endgroup$
    – user69323
    Jan 20, 2020 at 7:58
  • 2
    $\begingroup$ I guess this is a fix: s0 = Power[2, (3)^-1] (-(1/2) + (I Sqrt[3])/2); a0 = Power[2, (3)^-1]; mpq = MinimalPolynomial[s0, x, Extension -> ToNumberField[{s0, a0}][[1, 1]]]; mpcandidates = FactorList[mpq, Extension -> a0]; Select[mpcandidates[[All, 1]], Simplify[# == 0 /. x -> s0] &] $\endgroup$
    – Michael E2
    Jan 20, 2020 at 17:59
  • $\begingroup$ This method is too complex. MMA should redefine and encapsulate this function. $\endgroup$ Jan 21, 2020 at 1:30

1 Answer 1

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This is because MinimalPolynomial[s, x, Extension -> a] gives the characteristic polynomial of the algebraic number $s$ over the field $Q[a]$.

https://planetmath.org/characteristicpolynomialofalgebraicnumber

MinimalPolynomial[s, x, Extension ->a ] is equal to MinimalPolynomial[s, x]^d, where $d$ is the extension degree of $Q[a]$ over $Q[s]$. This requires that $s$ belong to $Q[a]$.

Unfortunately, the documentation page of MinimalPolynomial says differently. The following tutorial gives a correct specification:

https://reference.wolfram.com/language/tutorial/AlgebraicNumberFields.html

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