@Markiyan Hirnyk  You're right of course. Think of it as a joke!

@Markiyan Hirnyk 

P:=a*b^2+sqrt(41)*b*c-a^3*c^2;

P1:=subs(sqrt(41)=d, a*b^2+sqrt(41)*b*c-a^3*c^2);

Q:=expand(P1^7);

ind:=indets(%) minus {d};

n:=igcd(seq(degree(Q,s), s=ind));

subs(d=sqrt(41), RealDomain[simplify](Q^(1/n), symbolic));

@Markiyan Hirnyk 

1) I'm not familiar enough with the theory of polynomial ideals and radicals in them. As I understand it from the original post, it is necessary for a given polynomial in several variables over real numbers just to find the root from it of maximum order, of course, if this root there exists.

2)  I did not compare my method with other methods of solving this problem.

3) I just answered that question OP   "Is there another commands for this?"

@MDD  This means that an assumption is imposed on the variable a. If you want the symbol tilda to be absent, click

Tools->Options->Display->Assumed variables->No Annotation-> Apply Globally

@MDD  You must (in the copiable form) provide the full text of the code in which this error occurs.

@MDD   it is.

@Markiyan Hirnyk 

A := [x, y, x^2*y, x*y^2, y^2]:

B := [x^2, y^3]:

remove(c->`or`(seq(divide(c, b), b = B)) , A);

                                [x, y, x*y^2, y^2]

@Carl Love  Thank you very much for your response. This is a very effective way!

@Markiyan Hirnyk   for your interest! I meant that a segment is a closed interval. Of course easily to adjust procedure for all cases.

@Carl Love  I do not understand why on your images  additional lines appears, for example for k=1 .  Compare with my image:

F:=(x,y)->(1/2*x^2-3/2*y^3,-1/2*x^3+1/3*y^2):  # Nonlinear mapping R^2->R^2

X1:=t->(t,0): X2:=t->(0,t): X3:=t->(t,1): X4:=t->(1,t):

plot([[X1(t),t=0..1], [X2(t),t=0..1], [X3(t),t=0..1], [X4(t),t=0..1], [F(X1(t)), t=0..1], [F(X2(t)), t=0..1], [F(X3(t)), t=0..1], [F(X4(t)), t=0..1]], color=[red$4,blue$4], thickness=3, scaling=constrained);

@MDD  Your procedure was not working properly. What does  lt(p, T)  mean in it? An infinite loop occurs.

@shadi1386  Provide your data in Maple, rather than in Excel. I do not work with Excel.

@Markiyan Hirnyk  for useful information. I have Windows 8.1 64-bit

@rlopez  Should be  ctrl+l .

ctrl+L  does not work.

@farzane   

interface(rtablesize=infinity):

LinearAlgebra[RandomMatrix](8, 12);

convert({$1..12} minus {1,2,10,11}, list);

%%(..,%);