[Verse 1] Polynomial wandering through the field tonight Seeking roots that multiply and hide But in this basic ground it can't divide Into perfect linear pieces, smooth and bright The splitting field's the smallest place to go Where every factor breaks down nice and clean No more complexity left to be seen Just linear terms in one neat perfect row [Chorus] Split it down, split it clean Find the field where roots convene Galois group will map the way Automorphisms come to play Fix the base, but twist the rest Symmetries that pass the test Split it down, split it clean Mathematics unforeseen [Verse 2] Take Q with square root two adjoined The Galois group has just two moves Identity that nothing proves And sigma flipping signs combined Square root two becomes its twin Negative square root two instead While rationals stay safe in bed The cyclic group of two begins [Chorus] Split it down, split it clean Find the field where roots convene Galois group will map the way Automorphisms come to play Fix the base, but twist the rest Symmetries that pass the test Split it down, split it clean Mathematics unforeseen [Bridge] Primitive roots of unity dance Zeta sub n spins around Two pi i over n compound Euler's units get their chance Group of units mod n takes the stage Every coprime gets to move While the base field can't improve Galois theory writes each page [Verse 3] Field automorphisms guard the tower Keeping base points locked in place While extensions shift through space Revealing algebra's hidden power Smallest field that holds them all Where polynomials surrender whole Each root finds its destined role Linear factors standing tall [Outro] When the splitting's finally done Galois groups reveal the code Symmetries along the road Field extensions, one by one
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