(Subscript[p, 1]) = x + y + z - 6;
(Subscript[p, 2]) = (x^2) + (y^2) + (z^2) - 14;
(Subscript[p, 3]) = (x^4) + (y^4) + (z^4) - 98;

(Subscript[q, 2]) = Resultant[(Subscript[p, 1]), (Subscript[p, 2]), x]

(Subscript[q, 3]) = Resultant[(Subscript[p, 1]), (Subscript[p, 3]), x]

(Subscript[r, 3]) = Resultant[(Subscript[q, 2]), (Subscript[q, 3]), y]

Factor[(Subscript[r, 3])]

Factor[(Subscript[q, 2]) /. {z -> 1}]

Factor[(Subscript[q, 2]) /. {z -> 2}]

Factor[(Subscript[q, 2]) /. {z -> 3}]

Factor[(Subscript[p, 1]) /. {y -> 2, z -> 1}]

Factor[(Subscript[p, 1]) /. {y -> 3, z -> 1}]

Solve[{(Subscript[p, 1]) == 0, (Subscript[p, 2]) == 0, (Subscript[p, 3]) == 0}, {x, y, 
        z}]

(Subscript[p, 1]) = (x^3) + (y^2) - 2;
(Subscript[p, 2]) = (((x - y))^2) - 3;

f = Factor[Resultant[(Subscript[p, 1]), (Subscript[p, 2]), y]]

r = Root[f /. {x -> #} &, 1];

Factor[(Subscript[p, 1]) /. {x -> r}]

Factor[(Subscript[p, 1]) /. {x -> r}, 
        Extension -> {r}] /. {r ->  α}

Factor[(Subscript[p, 2]) /. {x -> r}, 
        Extension -> {r}] /. {r ->  α}

Solve[{(Subscript[p, 1]) == 0, (Subscript[p, 2]) == 0}, {x, y}]

Übungsaufgaben

FixedPointGraph[f_, {x_, x1_, x2_}, x0_, n_, options___] := 
      Module[{list, table}, 
        list = FixedPointList[N[(f /. x -> #)] &, x0, 
            n]; 
table = 
          Graphics[{Thickness[ .001], 
              Table[Line[{{list[[j]], list[[j]]}, {list[[j]], 
                      list[[j + 1]]}, {list[[j + 1]], 
                      list[[j + 1]]}}], {j, 1, 
                  Length[list] - 1}]}]; 
plot = 
          Plot[{f, x}, {x, x1, x2}, 
            DisplayFunction -> Identity]; 
Show[plot, 
          table, DisplayFunction -> $DisplayFunction, 
          options]]

FixedPointGraph[f_, {x_, x1_, x2_}, x0_] := 
      Module[{list, table}, 
        list = FixedPointList[N[(f /. x -> #)] &, 
            x0]; 
table = 
          Graphics[{Thickness[ .001], 
              Table[Line[{{list[[j]], list[[j]]}, {list[[j]], 
                      list[[j + 1]]}, {list[[j + 1]], 
                      list[[j + 1]]}}], {j, 1, 
                  Length[list] - 1}]}]; 
plot = 
          Plot[{f, x}, {x, x1, x2}, 
            DisplayFunction -> Identity]; 
Show[plot, 
          table, DisplayFunction -> $DisplayFunction]]