[Verse 1] In Galois land where fields extend There's a secret bridge that won't bend Take your group G and field L over K Watch the magic unfold this way Subgroups dancing with intermediate spaces Perfect partners in mathematical places [Chorus] Flip the mirror, what goes up comes down Bigger subgroups make smaller ground H maps to L fixed by every hand M maps back to Gal-land Index equals order, size equals height Two worlds spinning in opposite light [Verse 2] Start with subgroup H inside your G Find the elements that won't move free Sigma acts but leaves them still L to the H is the fixed field's thrill Count the automorphisms, count the degree Extension index matches perfectly [Chorus] Flip the mirror, what goes up comes down Bigger subgroups make smaller ground H maps to L fixed by every hand M maps back to Gal-land Index equals order, size equals height Two worlds spinning in opposite light [Bridge] When H sits normal in your G Then L fixed over K runs free Galois extension blooms like spring Normal subgroups are everything Field quotients match group quotients too The correspondence pushes through [Verse 3] Intermediate field between K and L Maps to automorphs that know it well Fixing group preserves what matters most Field and group are mirror hosts Inclusion reverses, big becomes small The fundamental truth connects them all [Outro] Group theory whispers field theory's name Two different languages, one single game Galois built the bridge between The most beautiful math you've ever seen
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