CODEWITHSUNDEEP
commandpolynomialsmaple
A. Brik
A. Brik

Reputation: 1

How to Convert RootOf Placeholder to Explicit Radicals in Maple for a Sextic Polynomial?

I have a sextic polynomial in Maple, and the solutions are displayed using RootOf. I want to convert these to explicit radicals if possible.

Here is the polynomial

49 Z^6 - 588 Z^5 + 2638 Z^4 - 5536 Z^3 + 5572 Z^2 - 2480 Z + 368

I tried several commands to extract the roots in radicals and did not work. I did solve it though using fsolve and obtained 6 positive numerical roots.

I used the galois command to determine its solvability by radicals and got this output:

"6T13", {"F_36(6):2", "S(3) wr 2", "[S(3)^2]2"}, "-", 72, {"(2 4 6)", "(2 4)", "(3 6)(1 4)(2 5)"}

What does this five elements mean?

Upvotes: 0

Views: 68

Answers (1)

dharr
dharr

Reputation: 126

Yes, Maple is unable to give an explicit solution to this polynomial. If p is your polynomial then neither of the following usual methods work.

convert(RootOf(p), radical)

solve(p, explicit, tryhard)

As for the output of galois, the most useful outputs are the last two: 72 is the order of the Galois group, and {"(2 4 6)", "(2 4)", "(3 6)(1 4)(2 5)"} are the generators of the Galois group as a permutation group. The others are more specific information on the nature of the group, as described on the help pages ?galois (online: https://www.maplesoft.com/support/help/Maple/view.aspx?path=galois) and ?group,transnames (https://www.maplesoft.com/support/help/Maple/view.aspx?path=group/transnames).

However, galois is an older command, and similar information can be obtained by the following sequence of newer commands:

with(GroupTheory):

Gr:=GaloisGroup(p,Z);

GroupOrder(Gr);

Generators(Gr);

IsSolvable(Gr);

which gives the same group order and generators and shows that the group is solvable. But now you have the actual group, and so can do things like find subgroups, etc.

However, my understanding of Galois groups is that this doesn't easily lead the the radical solutions themselves - see

https://math.stackexchange.com/questions/1469425/find-roots-or-splitting-field-of-a-polynomial-given-its-galois-group

or

https://math.stackexchange.com/questions/97587/solution-of-a-polynomial-of-degree-n-with-soluble-galois-group/97597#97597

Upvotes: 0

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