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[Axiom-developer] [Q] radicalSolve fails to find all roots ?
From: |
Vladimir Bondarenko |
Subject: |
[Axiom-developer] [Q] radicalSolve fails to find all roots ? |
Date: |
Mon, 17 Jan 2005 07:22:48 +0200 |
Hi *,
Any comments are highly appreciated on the following stuff.
Thank you in advance.
.....................................................................
Obviously, all the roots of the equation z^7 = 1 can be expressed in
radicals, and Mathematica can easily produce the explicit expressions
in terms of radicals.
Solve[z^7 == 1, z]
{{z -> 1}, {z -> -(-1)^(1/7)}, {z -> (-1)^(2/7)}, {z -> -(-1)^(3/7)},
{{z -> {z -> (-1)^(4/7)}, {z -> -(-1)^(5/7)}, {z -> (-1)^(6/7)}}
To save the space, below the only example is given.
FunctionExpand[ComplexExpand[-(-1)^(1/7)]]
(1/2)*((1/3)*((1/2)*(-1 + I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 +
I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) +
(1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6 + (3/4)*(-1 + I*Sqrt[3])*(-1 +
I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 +
I*Sqrt[3])^2))^(1/3)) + (1/4)*(-1 + I*Sqrt[3])^2*(6 + (3/4)*(-1 +
I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 + (3/4)*(-1 +
I*Sqrt[3])^2))^(1/3)) +(1/3)*((1/2)*(1 + I*Sqrt[7]) - ((-1 +
I*Sqrt[3])^2*((1/2)*(-1 -I*Sqrt[7]) + (1/2)*(-1 +
I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6
+ (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1
+ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) -(1/2)*(-1 + I*Sqrt[3])*(6 +
(3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 +
(3/4)*(-1 + I*Sqrt[3])^2))^(1/3))) + (1/2)*((1/3)*((1/2)*(-1 +
I*Sqrt[7]) + ((-1 + I*Sqrt[3])*((1/2)*(-1 + I*Sqrt[7]) + (1/2)*(-1 -
I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(2*(6
+ (3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1
+ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/4)*(-1 + I*Sqrt[3])^2*(6 +
(3/4)*(-1 + I*Sqrt[3])*(-1 + I*Sqrt[7]) + (1/2)*(-1 - I*Sqrt[7])*(1 +
(3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) + (1/3)*((1/2)*(-1 - I*Sqrt[7])
+((-1 + I*Sqrt[3])^2*((1/2)*(-1 - I*Sqrt[7]) + (1/2)*(-1 +
I*Sqrt[7])*((1/2)*(-1 + I*Sqrt[3]) + (1/4)*(-1 + I*Sqrt[3])^2)))/(4*(6
+ (3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1
+ (3/4)*(-1 + I*Sqrt[3])^2))^(1/3)) +(1/2)*(-1 + I*Sqrt[3])*(6 +
(3/4)*(-1 + I*Sqrt[3])*(-1 - I*Sqrt[7]) + (1/2)*(-1 + I*Sqrt[7])*(1 +
(3/4)*(-1 + I*Sqrt[3])^2))^(1/3)))
According to the AXIOM Book
AXIOM Book> Use radicalSolve if you want your solutions expressed in
AXIOM Book> terms of radicals.
However, already for z^7 = 1 this is not so,
-> radicalSolve(z^7=1, z)
[z= 1]
and the problem exists for 11, 13, 14, 15, 17, 19 etc
-> for i in 1..20 repeat print([i,#radicalSolve(z^i=1,z)])
[1,1]
[2,2]
[3,3]
[4,4]
[5,5]
[6,6]
[7,1] <-- not good
[8,8]
[9,9]
[10,10]
[11,1] <-- not good
[12,12]
[13,1] <-- not good
[14,2] <-- not good
[15,7] <-- not good
[16,16]
[17,1] <-- not good
[18,18]
[19,1] <-- not good
[20,20]
.....................................................................
Best,
Vladimir
- [Axiom-developer] [Q] radicalSolve fails to find all roots ?,
Vladimir Bondarenko <=