[Verse 1] When polynomials hide their secrets deep Four operations dance but roots still weep Addition, subtraction, multiply, divide Plus radical extraction as our guide Galois whispered truths we need to know The group that guards determines if roots show [Chorus] Solvable by radicals, the ancient quest When Galois groups are tame, we find the rest Normal subgroups stacking like a tower tall Each quotient must be abelian through it all S-one through S-four unlock every door But S-five breaks the pattern, blocks us from more [Verse 2] Symmetric groups hold polynomials' fate Four degrees submit, but five won't capitulate A chain of normal subgroups must unfold Where every factor's commutative and controlled But alternating five stands proud and simple No normal subgroups pierce its perfect dimple [Chorus] Solvable by radicals, the ancient quest When Galois groups are tame, we find the rest Normal subgroups stacking like a tower tall Each quotient must be abelian through it all S-one through S-four unlock every door But S-five breaks the pattern, blocks us from more [Bridge] It's not that we lack cleverness or skill The quintic's symmetry defies our will A-five lives inside S-five's heart Simple and non-abelian, tears hope apart Structure blocks extraction, not our minds The group itself determines what math finds [Outro] Radical solutions bow to group control Galois showed us symmetry's true role When groups unravel nice and neat Polynomials surrender at our feet But complex symmetries guard their treasure tight Some equations stay beyond our radical sight
← 3 The Fundamental Theorem of Galois Theory | 5 Constructibility →