How to tell if a polynomial has exact trigonometric or logarithmic roots?

The polynomial $$64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0$$ has seven roots, x = {1, $-\dfrac{1}{2}, \cos \dfrac{2n\pi}{11}$}, where n={1,2,3,4,5}.

Is there any way to tell if an arbitrary rational polynomial has any exact trigonometric or logarithmic (or whatever-ic) roots like this? Including complex roots? Is there an efficient way to find those roots if they exist? Sooo many problems I work on would get sooo much simpler if my approximate roots could be expressed exactly.

Here are a couple of polynomials with complex roots that I really wish I had exact expressions for:

$$p_1 \equiv 1 + 2 c^2 + 3 c^3 + 3 c^4 + 3 c^5 + c^6$$

$$p_2 \equiv 2 + 2 c + 4 c^2 + 6 c^3 + 6 c^4 + 6 c^5 + 4 c^6 + c^7$$

(...and here is a related Mathematica Question)

• Abel–Ruffini theorem. That's for the solutions in radicals though. Jan 25 '16 at 20:56
• @Kaster Ah, thank you. The AR thm seems to apply to radicals, not trigs or logs, but I appreciate a relevant and interesting pointer. Jan 25 '16 at 21:01
• Look up Chebyshev polynomials to see more polynomials with integer coefficients that have cosines of rational multiples of $\pi$ as their roots. Similar polynomials exist with sines, tangents, cotangents as roots. So on the trig side such polynomials exist, but you need to know what to look for. The values of logarithms are usually transcendental numbers meaning that they are not zeros of polynomials with integer coefficients. Jan 25 '16 at 21:15
• See for example here or here for more local examples. Jan 25 '16 at 21:19
• Anyway, unless I made a mistake, your polynomials $p_1$ and $p_2$ both have the full symmetric group as their Galois group. Meaning that they won't have trig functions at rational multiples of $\pi$ as their roots. For in those cases the Galois group would be abelian. If you have never heard of Galois theory then the above was probably all Hebrew to you. Jan 25 '16 at 21:31