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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.

.....................................................................

Obviously, all the roots of the equation z^7 = 1 can be expressed in
radicals, and Mathematica can easily produce the explicit expressions

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) + ((-1 + I*Sqrt)*((1/2)*(-1 +
I*Sqrt) + (1/2)*(-1 - I*Sqrt)*((1/2)*(-1 + I*Sqrt) +
(1/4)*(-1 + I*Sqrt)^2)))/(2*(6 + (3/4)*(-1 + I*Sqrt)*(-1 +
I*Sqrt) + (1/2)*(-1 - I*Sqrt)*(1 + (3/4)*(-1 +
I*Sqrt)^2))^(1/3)) + (1/4)*(-1 + I*Sqrt)^2*(6 + (3/4)*(-1 +
I*Sqrt)*(-1 + I*Sqrt) + (1/2)*(-1 - I*Sqrt)*(1 + (3/4)*(-1 +
I*Sqrt)^2))^(1/3)) +(1/3)*((1/2)*(1 + I*Sqrt) - ((-1 +
I*Sqrt)^2*((1/2)*(-1 -I*Sqrt) + (1/2)*(-1 +
I*Sqrt)*((1/2)*(-1 + I*Sqrt) + (1/4)*(-1 + I*Sqrt)^2)))/(4*(6
+ (3/4)*(-1 + I*Sqrt)*(-1 - I*Sqrt) + (1/2)*(-1 + I*Sqrt)*(1
+ (3/4)*(-1 + I*Sqrt)^2))^(1/3)) -(1/2)*(-1 + I*Sqrt)*(6 +
(3/4)*(-1 + I*Sqrt)*(-1 - I*Sqrt) + (1/2)*(-1 + I*Sqrt)*(1 +
(3/4)*(-1 + I*Sqrt)^2))^(1/3))) + (1/2)*((1/3)*((1/2)*(-1 +
I*Sqrt) + ((-1 + I*Sqrt)*((1/2)*(-1 + I*Sqrt) + (1/2)*(-1 -
I*Sqrt)*((1/2)*(-1 + I*Sqrt) + (1/4)*(-1 + I*Sqrt)^2)))/(2*(6
+ (3/4)*(-1 + I*Sqrt)*(-1 + I*Sqrt) + (1/2)*(-1 - I*Sqrt)*(1
+ (3/4)*(-1 + I*Sqrt)^2))^(1/3)) +(1/4)*(-1 + I*Sqrt)^2*(6 +
(3/4)*(-1 + I*Sqrt)*(-1 + I*Sqrt) + (1/2)*(-1 - I*Sqrt)*(1 +
(3/4)*(-1 + I*Sqrt)^2))^(1/3)) + (1/3)*((1/2)*(-1 - I*Sqrt)
+((-1 + I*Sqrt)^2*((1/2)*(-1 - I*Sqrt) + (1/2)*(-1 +
I*Sqrt)*((1/2)*(-1 + I*Sqrt) + (1/4)*(-1 + I*Sqrt)^2)))/(4*(6
+ (3/4)*(-1 + I*Sqrt)*(-1 - I*Sqrt) + (1/2)*(-1 + I*Sqrt)*(1
+ (3/4)*(-1 + I*Sqrt)^2))^(1/3)) +(1/2)*(-1 + I*Sqrt)*(6 +
(3/4)*(-1 + I*Sqrt)*(-1 - I*Sqrt) + (1/2)*(-1 + I*Sqrt)*(1 +
(3/4)*(-1 + I*Sqrt)^2))^(1/3)))

According to the AXIOM Book

However, already for z^7 = 1 this is not so,

[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,