[Verse 1] Ancient Greeks with compass drawn Tried to double cubes at dawn But Galois whispered through the years "Check the field extension here" Alpha's constructible they say When degree's a power of two's way Rational field to alpha's home Must have dimensions two can own [Chorus] Power of two, power of two That's the key that sees you through Straightedge and compass in your hand But extensions must be planned Two raised up to some whole number Or your construction's going under Power of two, it's always true For what geometry can do [Verse 2] Cube root of two has degree three Three's not two to any spree Doubling cubes will never work Though mathematicians used to lurk Trisecting angles sounds so neat But cubic roots you'll always meet Degree three blocks your pathway clean Impossible is what it means [Chorus] Power of two, power of two That's the key that sees you through Straightedge and compass in your hand But extensions must be planned Two raised up to some whole number Or your construction's going under Power of two, it's always true For what geometry can do [Bridge] Pi's transcendental, infinite degree Squaring circles? Can't you see? Regular polygons have their rule Fermat primes are the crucial tool Two to k times Fermat primes distinct All other n-gons are extinct [Verse 3] When field degree climbs past two's reign Construction efforts are in vain But when the tower splits just right With two-fold jumps at every height Then compass spins and straightedge glides Through algebraic countryside [Chorus] Power of two, power of two That's the key that sees you through Straightedge and compass in your hand But extensions must be planned Two raised up to some whole number Or your construction's going under Power of two, it's always true For what geometry can do [Outro] Galois solved the ancient quest Power of two puts dreams to rest
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