code: def dom_eigenvects(A, l=Dummy("lambda")):
      charpoly = A.charpoly()
      rows, cols = A.shape
      domain = A.domain
      _, factors = dup_factor_list(charpoly, domain)
  
      rational_eigenvects = []
      algebraic_eigenvects = []
      for base, exp in factors:
          if len(base) == 2:
              field = domain
              eigenval = -base[1] / base[0]
  
              EE_items = [
                  [
                      eigenval if i == j else field.zero
                      for j in range(cols)
                  ]
                  for i in range(rows)
              ]
              EE = DomainMatrix(EE_items, (rows, cols), field)
  
              basis = (A - EE).nullspace(divide_last=True)
              rational_eigenvects.append((field, eigenval, exp, basis))
          else:
              minpoly = Poly.from_list(base, l, domain=domain)
              field = FiniteExtension(minpoly)
              eigenval = field(l)
  
              AA_items = [
                  [
                      Poly.from_list([item], l, domain=domain).rep
                      for item in row
                  ]
                  for row in A.rep.to_ddm()
              ]
              AA_items = [
                  [field(item) for item in row] for row in AA_items
              ]
              AA = DomainMatrix(AA_items, (rows, cols), field)
              EE_items = [
                  [
                      eigenval if i == j else field.zero
                      for j in range(cols)
                  ]
                  for i in range(rows)
              ]
              EE = DomainMatrix(EE_items, (rows, cols), field)
  
              basis = (AA - EE).nullspace(divide_last=True)
              algebraic_eigenvects.append((field, minpoly, exp, basis))
  
      return rational_eigenvects, algebraic_eigenvects
  
file_path: sympy/sympy/polys/matrices/eigen.py
imports: from sympy.core.symbol import Dummy
  from ..agca.extensions import FiniteExtension
  from ..factortools import dup_factor_list
  from ..polyroots import roots
  from ..polytools import Poly
  from ..rootoftools import CRootOf
  from .domainmatrix import DomainMatrix
label_desc: Use image node_id 1 for calling a global function with example usage: dom_eigenvects(A, l) and returns: rational_eigenvects,  algebraic_eigenvects