code: def minimal_polynomial(
      ex, x=None, compose=True, polys=False, domain=None
  ):
      """
      Computes the minimal polynomial of an algebraic element.
  
      Parameters
      ==========
  
      ex : Expr
          Element or expression whose minimal polynomial is to be calculated.
  
      x : Symbol, optional
          Independent variable of the minimal polynomial
  
      compose : boolean, optional (default=True)
          Method to use for computing minimal polynomial. If ``compose=True``
          (default) then ``_minpoly_compose`` is used, if ``compose=False`` then
          groebner bases are used.
  
      polys : boolean, optional (default=False)
          If ``True`` returns a ``Poly`` object else an ``Expr`` object.
  
      domain : Domain, optional
          Ground domain
  
      Notes
      =====
  
      By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
      are computed, then the arithmetic operations on them are performed using the resultant
      and factorization.
      If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
      The default algorithm stalls less frequently.
  
      If no ground domain is given, it will be generated automatically from the expression.
  
      Examples
      ========
  
      >>> from sympy import minimal_polynomial, sqrt, solve, QQ
      >>> from sympy.abc import x, y
  
      >>> minimal_polynomial(sqrt(2), x)
      x**2 - 2
      >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
      x - sqrt(2)
      >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
      x**4 - 10*x**2 + 1
      >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
      x**3 + x + 3
      >>> minimal_polynomial(sqrt(y), x)
      x**2 - y
  
      """
  
      ex = sympify(ex)
      if ex.is_number:
          # not sure if it's always needed but try it for numbers (issue 8354)
          ex = _mexpand(ex, recursive=True)
      for expr in preorder_traversal(ex):
          if expr.is_AlgebraicNumber:
              compose = False
              break
  
      if x is not None:
          x, cls = sympify(x), Poly
      else:
          x, cls = Dummy("x"), PurePoly
  
      if not domain:
          if ex.free_symbols:
              domain = FractionField(QQ, list(ex.free_symbols))
          else:
              domain = QQ
      if hasattr(domain, "symbols") and x in domain.symbols:
          raise GeneratorsError(
              "the variable %s is an element of the ground "
              "domain %s" % (x, domain)
          )
  
      if compose:
          result = _minpoly_compose(ex, x, domain)
          result = result.primitive()[1]
          c = result.coeff(x ** degree(result, x))
          if c.is_negative:
              result = expand_mul(-result)
          return (
              cls(result, x, field=True) if polys else result.collect(x)
          )
  
      if not domain.is_QQ:
          raise NotImplementedError("groebner method only works for QQ")
  
      result = _minpoly_groebner(ex, x, cls)
      return cls(result, x, field=True) if polys else result.collect(x)
  
imports: from functools import reduce
  from sympy.core.add import Add
  from sympy.core.exprtools import Factors
  from sympy.core.function import expand_mul, expand_multinomial, _mexpand
  from sympy.core.mul import Mul
  from sympy.core.numbers import I, Rational, pi, _illegal
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy
  from sympy.core.sympify import sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.functions.elementary.exponential import exp
  from sympy.functions.elementary.miscellaneous import sqrt, cbrt
  from sympy.functions.elementary.trigonometric import cos, sin, tan
  from sympy.ntheory.factor_ import divisors
  from sympy.utilities.iterables import subsets
  from sympy.polys.domains import ZZ, QQ, FractionField
  from sympy.polys.orthopolys import dup_chebyshevt
  from sympy.polys.polyerrors import NotAlgebraic, GeneratorsError
  from sympy.polys.polytools import Poly, PurePoly, invert, factor_list, groebner, resultant, degree, poly_from_expr, parallel_poly_from_expr, lcm
  from sympy.polys.polyutils import dict_from_expr, expr_from_dict
  from sympy.polys.ring_series import rs_compose_add
  from sympy.polys.rings import ring
  from sympy.polys.rootoftools import CRootOf
  from sympy.polys.specialpolys import cyclotomic_poly
  from sympy.utilities import numbered_symbols, public, sift
label_desc: Use image node_id 17 for calling a global function with example usage: minimal_polynomial(ex, x, compose, polys, domain) and returns: unknown, unknown