Unsolvable has three distinct roots – the separator is which tools were allowed.
One word, unsolvable, names at least three different things. This paper writes their addresses and shows that the separator is which tools were allowed. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Abel-Ruffini, Galois theory, Wantzel’s theorem, Gauss’s seventeen-gon and Matiyasevich’s theorem are all standard.
First, the obstruction is a derived series that stalls. The derived series of the symmetric groups on three and four letters reach the trivial group, at orders six, three, one and twenty-four, twelve, four, one. That of the symmetric group on five letters, however, stalls after dropping from one hundred and twenty to sixty (computed here). What it stalls on is the alternating group of order sixty, simple and non-abelian, and nothing further can be peeled away. Up to the quartic everything peels, so there is a formula; from the quintic it does not, so there is none.
Second, a brake. Unsolvable, and yet the root is computable. The polynomial x to the fifth minus x minus one is irreducible and its Galois group is the full symmetric group, so it has no expression in radicals. Its real root is nonetheless 1.16730397826141868425604589985, to as many digits as one asks (computed here at forty-digit precision, with a residual of order ten to the minus forty-one). Unsolvable therefore says only that no expression built from the four operations and radicals exists; it does not say the number is out of reach. Impossibility is not a property of the object but a relation between the object and the tools.
Third, another brake. The quintic cannot be solved is false as stated. The polynomial x to the fifth minus two is irreducible, yet its roots are all expressible in radicals, beginning with the fifth root of two; the cyclotomic case is similar. The true statement is that there is no general formula in radicals for the general quintic, and solvable quintics are plentiful. A claim about the general case must not be traded for a claim about a particular one – the same shape as brakes the corpus has applied before.
Fourth, change the toolbox and the obstruction changes. With straightedge and compass a constructible number must have degree a power of two (Wantzel 1837). From the triple-angle identity, twice the cosine of twenty degrees satisfies a cubic, and that cubic is irreducible over the rationals (checked here, with a numerical residual of order ten to the minus forty-one). The degree is therefore three, not a power of two, so the angle cannot be trisected; doubling the cube fails for the same reason, the cube root of two also having degree three. The regular seventeen-gon, by contrast, can be constructed, because Euler’s function at seventeen is sixteen, that is, two to the fourth (Gauss 1796). At seven and nine the value is six and at nineteen it is eighteen, and none of those can be constructed. With the same tool the heptagon fails and the seventeen-gon succeeds, and what separates them is a single number.
Fifth, a third toolbox: a finite procedure. Hilbert’s tenth problem asked whether there is an algorithm deciding if a given Diophantine equation has an integer solution. Matiyasevich (1970) answered no. Here unsolvable differs again from both earlier senses: no formula is missing and no construction is impossible, but there is no procedure at all. More digits will not help, and neither will more patience – quite unlike the quintic above, which came out to thirty digits.
The census. Not solvable in radicals means that, with the four operations and radicals allowed, the Galois group is not solvable. Not constructible means that, with straightedge and compass allowed, the degree is not a power of two. Undecidable means that, with a finite procedure allowed, there is no procedure. The separator is not how hard the problem is but which tools were allowed. Change the toolbox and the same object moves from impossible to routine: the quintic’s root lies beyond radicals and thirty digits from a computer.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: writing addresses for three senses of unsolvable, naming the separator as the toolbox, and fencing the widely repeated claim that the quintic cannot be solved, by counterexample. What remains: impossible is not a label stuck on an object. It is the name of a relation between the object and the tools one brought. To say unsolvable without naming the tools is to have said nothing yet.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). “Unsolvable” Has Three Distinct Roots [Paper 25]. Zenodo.
DOI: 10.5281/zenodo.22090297
ORCID: 0009-0006-8587-3450