code: def dup_euclidean_prs(f, g, K):
      """
      Euclidean polynomial remainder sequence (PRS) in `K[x]`.
  
      Examples
      ========
  
      >>> from sympy.polys import ring, QQ
      >>> R, x = ring("x", QQ)
  
      >>> f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
      >>> g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
  
      >>> prs = R.dup_euclidean_prs(f, g)
  
      >>> prs[0]
      x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
      >>> prs[1]
      3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
      >>> prs[2]
      -5/9*x**4 + 1/9*x**2 - 1/3
      >>> prs[3]
      -117/25*x**2 - 9*x + 441/25
      >>> prs[4]
      233150/19773*x - 102500/6591
      >>> prs[5]
      -1288744821/543589225
  
      """
      prs = [f, g]
      h = dup_rem(f, g, K)
  
      while h:
          prs.append(h)
          f, g = g, h
          h = dup_rem(f, g, K)
  
      return prs
  
imports: from sympy.polys.densearith import dup_sub_mul, dup_neg, dmp_neg, dmp_add, dmp_sub, dup_mul, dmp_mul, dmp_pow, dup_div, dmp_div, dup_rem, dup_quo, dmp_quo, dup_prem, dmp_prem, dup_mul_ground, dmp_mul_ground, dmp_mul_term, dup_quo_ground, dmp_quo_ground, dup_max_norm, dmp_max_norm
  from sympy.polys.densebasic import dup_strip, dmp_raise, dmp_zero, dmp_one, dmp_ground, dmp_one_p, dmp_zero_p, dmp_zeros, dup_degree, dmp_degree, dmp_degree_in, dup_LC, dmp_LC, dmp_ground_LC, dmp_multi_deflate, dmp_inflate, dup_convert, dmp_convert, dmp_apply_pairs
  from sympy.polys.densetools import dup_clear_denoms, dmp_clear_denoms, dup_diff, dmp_diff, dup_eval, dmp_eval, dmp_eval_in, dup_trunc, dmp_ground_trunc, dup_monic, dmp_ground_monic, dup_primitive, dmp_ground_primitive, dup_extract, dmp_ground_extract
  from sympy.polys.galoistools import gf_int, gf_crt
  from sympy.polys.polyconfig import query
  from sympy.polys.polyerrors import MultivariatePolynomialError, HeuristicGCDFailed, HomomorphismFailed, NotInvertible, DomainError
label_desc: Use image node_id 7 for calling a global function with example usage: dup_euclidean_prs(f, g, K) and returns: prs