code: def _minpoly_op_algebraic_element(
      op, ex1, ex2, x, dom, mp1=None, mp2=None
  ):
      """
      return the minimal polynomial for ``op(ex1, ex2)``
  
      Parameters
      ==========
  
      op : operation ``Add`` or ``Mul``
      ex1, ex2 : expressions for the algebraic elements
      x : indeterminate of the polynomials
      dom: ground domain
      mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None
  
      Examples
      ========
  
      >>> from sympy import sqrt, Add, Mul, QQ
      >>> from sympy.polys.numberfields.minpoly import _minpoly_op_algebraic_element
      >>> from sympy.abc import x, y
      >>> p1 = sqrt(sqrt(2) + 1)
      >>> p2 = sqrt(sqrt(2) - 1)
      >>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ)
      x - 1
      >>> q1 = sqrt(y)
      >>> q2 = 1 / y
      >>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y))
      x**2*y**2 - 2*x*y - y**3 + 1
  
      References
      ==========
  
      .. [1] https://en.wikipedia.org/wiki/Resultant
      .. [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
             "Degrees of sums in a separable field extension".
  
      """
      y = Dummy(str(x))
      if mp1 is None:
          mp1 = _minpoly_compose(ex1, x, dom)
      if mp2 is None:
          mp2 = _minpoly_compose(ex2, y, dom)
      else:
          mp2 = mp2.subs({x: y})
  
      if op is Add:
          # mp1a = mp1.subs({x: x - y})
          if dom == QQ:
              R, X = ring("X", QQ)
              p1 = R(dict_from_expr(mp1)[0])
              p2 = R(dict_from_expr(mp2)[0])
          else:
              (p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y)
              r = p1.compose(p2)
              mp1a = r.as_expr()
  
      elif op is Mul:
          mp1a = _muly(mp1, x, y)
      else:
          raise NotImplementedError("option not available")
  
      if op is Mul or dom != QQ:
          r = resultant(mp1a, mp2, gens=[y, x])
      else:
          r = rs_compose_add(p1, p2)
          r = expr_from_dict(r.as_expr_dict(), x)
  
      deg1 = degree(mp1, x)
      deg2 = degree(mp2, y)
      if op is Mul and deg1 == 1 or deg2 == 1:
          # if deg1 = 1, then mp1 = x - a; mp1a = x - y - a;
          # r = mp2(x - a), so that `r` is irreducible
          return r
  
      r = Poly(r, x, domain=dom)
      _, factors = r.factor_list()
      res = _choose_factor(factors, x, op(ex1, ex2), dom)
      return res.as_expr()
  
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 5 for calling a global function with example usage: _minpoly_op_algebraic_element(op, ex1, ex2, x, dom, mp1, mp2) and returns: res, r