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.
Best Answer
After more thought, I feel sure enough to claim in an answer that $1^{1/5}$ is not a "solution in radicals" to $x^5-1=0$ in the sense that is guaranteed to exist by the solvability of the equation's Galois group. I think the kind of solution that the solvability of the Galois group implies exists is exactly a "non-trivial" solution in your sense.
The "solution in radicals" that is guaranteed by the solvability of the Galois group of a polynomial $f$ is really a "root tower" over $\mathbb{Q}$: a tower of fields, beginning with $\mathbb{Q}$ and ending with a field containing $f$'s splitting field, in which each field is obtained from the last one by adjoining a $p$th root for some prime $p$.
$$\mathbb{Q}\subset\mathbb{Q}[r_1]\subset\dots\subset\mathbb{Q}[r_1,\dots,r_k]$$
where for each $r_j$ there is a prime $p_j$ such that $r_j^{p_j}$ was already in the previous field $\mathbb{Q}[r_1,\dots,r_{j-1}]$ but $r_j$ is not. The mechanism of the proof is that if the Galois group $G$ is solvable, then there is a composition series
$$G=G_0 \triangleright G_1\triangleright \dots \triangleright G_k=\{1\}$$
such that each factor group $G_{j-1}/G_j$ is cyclic of prime order $p_j$. By the fundamental theorem of Galois theory, there is thus a tower of fields
$$\mathbb{Q}=K_0 \subset K_1 \subset \dots \subset K_k=\mathrm{splittingfield}(f)$$
where $K_j$ has degree $p_j$ over $K_{j-1}$. By possibly adjoining some extra elements, it can be guaranteed that each field extension can be accomplished by adjoining a $p_j$th root to the previous field (in other words a root $r_j$ of the polynomial $x^{p_j}-a$ where $a$ is in the previous field), so that the tower has the above form (a "root tower"). Because extra elements may have been added, the final field may now be bigger than $f$'s splitting field. (I am sweeping some possibly pertinent details under the rug here: the extra elements you need to adjoin are precisely the $p_j$th roots of unity. The argument, when applied to cyclotomic polynomials, is saved from circularity by induction on the size of the primes.)
The key point is this: when $p_j$th roots are adjoined, the field extensions have degree $p_j$. This means that the polynomial $x^{p_j}-a$ of which $r_j$ is a root must be irreducible over $K_{j-1}$. If the polynomial were reducible, $r_j$ would be the root of a polynomial of degree lower than $p_j$ and $K_j=K_{j-1}[r_j]$ would have degree less than $p_j$ over $K_{j-1}$.
What I'm getting at is that the $p$th roots that are adjoined at each step in such a "solution by radicals" are all roots of polynomials $x^p-a$ that are irreducible over the previous field. In particular, $$ \mathbb{Q} \subset \mathbb{Q}[1^{1/5}]$$ is not a solution of $x^5-1$ by radicals in this sense, because $x^5-1$ isn't irreducible over $\mathbb{Q}$. In fact, in this context as you note, $1^{1/5}$ doesn't even have an unambiguous meaning. It could be $1$, in which case this is not even a nontrivial field extension. On the other hand it could be any of the four primitive fifth roots of unity, and then the expression $\mathbb{Q}[1^{1/5}]$ would have a meaning determined up to isomorphism, but it still wouldn't be a "solution by radicals" because fifth roots of unity do not solve an irreducible polynomial of the form $x^p-a$.
Furthermore, the solutions to an equation furnished by a "solution by radicals" of this kind are always "non-trivial" in your sense. Since at each stage the adjoined root $r_j$ is a root of an irreducible polynomial over $K_{j-1}$, replacing it with any other root of this polynomial induces an automorphism of $K_j$ fixing $K_{j-1}$ pointwise. At the end of everything, the roots of $f$ (the original polynomial to be solved) are elements of the top field $\mathbb{Q}[r_1,\dots,r_k]$; thus they are rational expressions in $r_1,\dots,r_k$ with rational coefficients. Replacing any of the $r_j$'s with a different root of the same irreducible polynomial it was a root of thus induces an automorphism of the top field $\mathbb{Q}[r_1,\dots,r_k]$ that fixes $\mathbb{Q}$ pointwise. Thus any expression in $r_1,\dots,r_k$ that solves $f$ will still solve $f$ after this replacement.
To make the connection to the question explicit, this means that the cyclotomic polynomials, because they have solvable Galois groups, do all have legitimate / "non-trivial" solutions in radicals, and this result dates back to the days of Gauss and Galois.