Four dials hold the four roots of x⁴ − 2. The lock only turns in ways that keep every algebraic relation on its lid true. Find out which arrangements it will ever reach.
The roots. x⁴ − 2 has four roots, α, iα, −α, −iα, with α = ⁴√2. A symmetry of the roots is a rearrangement that respects every polynomial relation with rational coefficients. Those rearrangements form the Galois group.
The lid. Three relations are enough to pin the group down: dial 1 + dial 3 = 0, dial 2 + dial 4 = 0, and (dial 1 · dial 2)² = −2. The lock refuses any move that would make one of them false.
What it allows. "Turn" sends α to iα and fixes i, so it rotates the square a quarter turn. "Flip" is complex conjugation, a reflection of the square. Together they generate the dihedral group D₄, the eight symmetries of a square. That is the Galois group of x⁴ − 2 over ℚ.
What it forbids. Of the 24 ways to arrange four roots, only 8 open. Swapping α with iα alone breaks dial 1 + dial 3 = 0, since iα + (−α) ≠ 0. When a target is not one of the 8, the right answer is "impossible."
Why it matters. The group's structure (it has a normal series with abelian steps) is exactly why x⁴ − 2 can be solved with radicals. A polynomial whose group is all of S₅ cannot.