I am seeking algebraic expressions which solve a polynomial equation, in particular an arbitrary cyclotomic polynomial. Let us agree we are not talking about expressions such as $e^{2\pi/7}$. My problem is to distinguish between those expressions I am going to call "trivial" and "non-trivial".
Typically, algebraic expressions with radical signs admit of multiple choices for the value of the radical expression: two for a square root, three for a cube root, etc. I will define a "non-trivial" algebraic solution as an expression in radicals such that for any choice of values, the expression is always a solution of the equation. Example: for any quadratic equation, there are two choices for the square root, and both choices give correct solutions.
For the fifth cyclotomic polynomial, the solution (which I looked up) is given by the expression: \[ \frac{\sqrt5 - 1}4 + \frac{\sqrt{-5 - \sqrt5}}8 \] This expression takes on four possible values, and each of them is in fact a true solution of the fifth cyclotomic polynomial. It might look like it takes on eight values because there are three choices of the square root, but two of them must be the same choice: you cannot take one value for the square root of five, and then later in the same expression switch to the other value.
On the other hand, we have an alternative solution for the same polynomial: \[ 1^{1/5} \] I define this as a "trivial" solution because it takes on five possible values, but only four of them are true solutions of the fifth cyclotomic polynomial. This expression is, of course, a non-trivial solution of the non-irreducible polynomial $x^5 - 1 = 0$, but that's not the polynomial we're talking about.
I recently posted a similar question, which I believe was correctly answered by Paul Garret, but I am not at all sure that everyone who participated in the discussion was actually working on the same question. I hope this clarifies the situation.
My question, then, was and is: do all the cyclotomic polynomials have "non-trivial" solutions? If not, what is the smallest polynomial for which such a solution (a) is not known, or (b) can be shown not to exist? I cannot believe this question is nonsense, and I would be very surprised if it hasn't been answered in the literature.
EDIT I am extremely gratifying that you guys had some fun with my question. I have always loved this topic and I think it is the greatest miracle of all math that the six choices of the cubic formula (three cube roots and two square roots) can be made to logically and unambiguously collapse to exactly three values.
Because I have only one check mark to award for "accepted answer", from many deserving contributors, I award my coveted check-mark to Ben. Thanks again guys.