De Moivre's Quintic

Author

Eric W. Weisstein
October 1, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/deMoivresQuintic.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/deMoivresQuintic.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

deMoivre=x^5+a x^3+a^2/5x+b;

Roots

soln=Solve[deMoivre==0,x]

Table[deMoivre/.soln/.Thread[{a,b}->Table[Random[Complex,{-1,1}5(I+1)],{2}]],{10}]//Chop

Table[deMoivre/.soln/.Thread[{a,b}->Table[Random[Real,{-1,1}5],{2}]],{10}]//Chop

Solution

us=ExpandAll[Solve[{u1+u2==-b,u1 u2==-(a/5)^5},{u1,u2}]]

Expand[{u1+u2,u1 u2}/.us]

soln1=With[{ω=Exp[2Pi I/5]},Table[(ω^j) (u1^(1/5))+(ω^(4j))(u2^(1/5)),{j,0,4}]]/.us[[1]]

soln2=With[{ω=Exp[2Pi I/5]},Table[(ω^j) (u1^(1/5))+(ω^(4j))(u2^(1/5)),{j,0,4}]]/.us[[2]]

Something is Wrong; doesn't check

Sort[x/.soln/.{a->.1,b->.2}]

Sort[soln1/.{a->.1,b->.2}]

Sort[soln2/.{a->.1,b->.2}]

deMoivre/.x->soln1/.{a->.1,b->.2}

deMoivre/.x->soln2/.{a->.1,b->.2}