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## [Axiom-developer] [#12 radicalSolve fails to find all roots ?]

 From: Bill Page Subject: [Axiom-developer] [#12 radicalSolve fails to find all roots ?] Date: Mon, 17 Jan 2005 22:48:17 -0600

??changed:
-These are NOT bugs! But the following may be! Consider the equation z^n=1 for n
-= 7:
-
-
-   (1)
-       7+-+      +---+7+-+    2%pi    7+-+    2%pi
-   [z= \|2 , z= \|- 1 \|2 sin(----) + \|2 cos(----),
-                                7               7
-        +---+7+-+    4%pi    7+-+    4%pi
-    z= \|- 1 \|2 sin(----) + \|2 cos(----),
-                       7               7
-        +---+7+-+    6%pi    7+-+    6%pi
-    z= \|- 1 \|2 sin(----) + \|2 cos(----),
-                       7               7
-        +---+7+-+    8%pi    7+-+    8%pi
-    z= \|- 1 \|2 sin(----) + \|2 cos(----),
-                       7               7
-        +---+7+-+    10%pi    7+-+    10%pi
-[9 more lines...]
These are NOT bugs! But the following may be! Consider the equation $z^n=1$ for
$n=7$
\begin{axiom}
\end{axiom}

Of course, these are correct solutions by Euler's Formula. A bit surprising
that
radicalSolve invokes these for $z^7=2$ and not for $z^7=1$; when $n$ is 7, these

??changed:
-http://mathworld.wolfram.com/ConstructiblePolygon.html
-http://mathworld.wolfram.com/TrigonometryAngles.html
-  http://mathworld.wolfram.com/ConstructiblePolygon.html

-  http://mathworld.wolfram.com/TrigonometryAngles.html

??changed:
-p. 34).
-
-n =  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17 18 19 20
-phi= 1  1  2  2  4  2  6  4  6   4  10   4  12   6   8   8  16  6 18  8
-bad=                   x     x       x       x   x              x  x
-n =                    7            11      13  14  15      17    19
p. 34)::

n =  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17 18 19 20
phi= 1  1  2  2  4  2  6  4  6   4  10   4  12   6   8   8  16  6 18  8
bad=                   x     x       x       x   x              x  x
n =                    7            11      13  14  15      17    19

??changed:
-in terms of radicals and arithmetic operations. It did not find those for n =
15
-and 17 probably (I am guessing) because at the time of implementation, these
in terms of radicals and arithmetic operations. It did not find those for $n = 15$
and $17$ probably (I am guessing) because at the time of implementation, these

??changed:
-for n = 9, 18, the solutions are expressible in radicals only if radicals of
for $n = 9, 18$, the solutions are expressible in radicals only if radicals of

??changed:
expansion for $(-1)^{1/7}$ that Vladimir gave involves radicals of complex

??changed:
-is why I am surprised at the above result for z^7=2).
-
-In other words, rather than viewing the answer for z^7=1 as a bug, we should
-view the answers for z^7=2, z^7=3 (and may be even z^9=1, z^18=1) as bugs!
is why I am surprised at the above result for $z^7=2$).

In other words, rather than viewing the answer for $z^7=1$ as a bug, we should
view the answers for $z^7=2$, $z^7=3$ (and may be even $z^9=1$, $z^{18}=1$) as
bugs!

??changed:
-
-   (1)
-                                             +------------+
-                                             |   +---+
-        +------------+       +---+ +-+       |- \|- 3  - 1
-        |   +---+          (\|- 1 \|3  - 1) 3|------------
-        |- \|- 3  - 1                       \|      2
-   [z= 3|------------ , z= -------------------------------,
-       \|      2                          2
-                           +------------+
-                           |   +---+
-           +---+ +-+       |- \|- 3  - 1      +----------+
-       (- \|- 1 \|3  - 1) 3|------------      | +---+
-                          \|      2           |\|- 3  - 1
-    z= ---------------------------------, z= 3|---------- ,
-                       2                     \|     2
-                         +----------+                        +----------+
-                         | +---+                             | +---+
-[60 more lines...]
\begin{axiom}
\end{axiom}

--removed:
------------------
-
->
-> 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)}}
-[88 more lines...]

??changed:
-Obviously, all the roots of the equation z^7 = 1 can be expressed in
Obviously, all the roots of the equation $z^7 = 1$ can be expressed in

??changed:
-
-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 +
-[23 more lines...]

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--

??changed:
-
-   [z= 1]
\begin{axiom}
\end{axiom}

??changed:
--> 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
-[4 more lines...]
\begin{axiom}
for i in 1..20 repeat print([i,#radicalSolve(z^i=1,z)])
\end{axiom}

::

[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,

--