Problem suggested by David McCooey

[Istvan Lauko]

I am relaying a problem from David

here is what he wrote:

About me:  In 1980, I went on a mathematics competition trip organized

by George Berzsenyi and the Texas ARML.  It was a very rewarding

experience, but I entered university in the fall and did not keep in

touch with George.  A couple weeks ago, I contacted George to see

if he'd be interested in a math problem I had been working on.

I don't know whether this problem has been solved already.

Here's the problem:

Find coordinates for a regular heptagon in 3D Euclidean space where

all 3 components (x,y,z) of all 7 coordinates are elements of the same

cubic field, or prove that it can't be done.

The problem can be stated equivalently, with more algebra, as follows:

Find a unit vector (x,y,z) that allows all 9 entries of the following rotation

matrix to be elements of the same cubic field:

(1-c)*x^2 + c       (1-c)*x*y - z*s     (1-c)*x*z + y*s

(1-c)*x*y + z*s     (1-c)*y^2 + c       (1-c)*y*z - x*s

(1-c)*x*z - y*s     (1-c)*y*z + x*s     (1-c)*z^2 + c

where:

c = cos(2*pi/7)

s = sin(2*pi/7)

or prove that no such vector exists.

When I contacted George, it had occurred to me that this problem was

similar to competition math problems, but the notion of a field is usually

beyond high school mathematics, at least in 1980 USA.  However, there

are always students with exceptional talent and knowledge, and one of

them may be able to solve this problem.

As I told George, I have not fully explored the number-theoretic aspects

of this problem, so a simple solution or disproof may yet exist.

I have been pursuing it in my off time, mostly using computer searches.

Best regards,

David McCooey

[David Ash]

I don't have a complete solution, but some thoughts:

By taking differences we determine that x*s and y*s must be in the field, and we similarly determine by taking sums that (1-c)*x*y must be in the field. Taking ratios, (1-c)/(s^2)= 1/(1+c) must be in the field, so c must be in the field. (If either x=0 or y=0, this would involve a division by zero, but in that case we can deduce more trivially that c must be in the field).

As c is a root of the cubic 8x^3+4x^2-4x-1=0, the field must be precisely Q[c] if a solution exists. As x*s, y*s, and z*s are all in the field, let x=X/s, y=Y/s, and z=Z/s where X,Y, and Z are all in the field. In this case it can be shown that all nine elements must be in the field. So a necessary and sufficient condition for (x,y,z) to be a solution is for x*s, y*s, and z*s to all be in the field.

In this case x^2+y^2+z^2=1 so X^2+Y^2+Z^2=s^2=1-c^2. So a necessary and sufficient condition for a solution to exist is to find X,Y,Z in Q[c] with X^2+Y^2+Z^2=1-c^2.

That's as far as I've gotten.