Why u is algebraic.

The existence question.

How the minimal polynomials were found.

Here are the minimal polynomials. So far as I know these were not previously known and have not been published elsewhere.

I was only successful in finding such polynomials for *n*<19
and for *n*=20, 24, 38, 48, 50, 52, 120.

*u*=cos(central angle to minimal edge). Thus, for example,
the optimal solution for Tammes' problem for *n*=10 has *u*
satisfying
the polynomial named
below.

Note:

The minimum polynomial factors over Q(sqrt(2)) when n=8, 16;

the minimum polynomial factors over Q(sqrt(3)) when n=38;

the minimum polynomial factors over Q(sqrt(5)) when n=11, 12, 120;

the minimum polynomial factors over Q(9th roots of unity) when n=7.

For these same polynomial in maple format click
here .