Expand cos(2*pi/n) for n=5, 17, 257, 65537 to Radicals (Fermat prime numbers)

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Using the method radical_expression for algebraic real field element, you can sometimes obtain a result. Its doc says Attempt to obtain a symbolic expression using radicals. If no exact symbolic expression can be found, the algebraic number will be returned without modification:

sage: AA
Algebraic Real Field
sage: AA(cos(2*pi/5)).radical_expression()
1/4*sqrt(5) - 1/4
sage: AA(cos(2*pi/7)).radical_expression()
(7/144*I*sqrt(3) + 7/432)^(1/3) + 7/36/(7/144*I*sqrt(3) + 7/432)^(1/3) - 1/6

But it does not work for n=17 and n=257:

sage: AA(cos(2*pi/17)).radical_expression()
0.9324722294043558?
sage: AA(cos(2*pi/257)).radical_expression()
0.9997011578430937?

Here in a sample code using GAP's RadiRoot package that works for $n=5$ and $n=17$. It produces the result in the form of a LaTeX file. For $n=257$, it gives an error Transitive groups of degree 128 are not available, while for $n=65537$ it goes into lengthy calculations (I was not patient enough to wait for an answer).

n = 5
f = cos(2*pi/n).minpoly()
x = f.variables()[0]
gap.eval('LoadPackage("radiroot");')
gap.eval(f'{x} := Indeterminate( Rationals, "{x}" );')
assert gap.eval(f'IsSolvablePolynomial({f});')     # necessary condition
fname = gap.eval(f'RootsOfPolynomialAsRadicals( {f}, "latex" );').strip('"')
with open(fname) as file:
    for line in file:
        print(fr'{line.rstrip()}')