[Verse 1] Fields inside fields like Russian dolls K subset L, the vector space calls Degree is dimension, count the base Complex over reals, just two in place One and i will span the whole Square root two makes rationals roll [Chorus] Field extensions, towers climb Algebraic or transcendental time Simple extensions, minimal degree Tower law multiplies what we see Finite fields with prime power size Perfect laboratories in disguise [Verse 2] Alpha's algebraic when polynomials bind Root of some equation we can find But pi and e transcend the scene No rational polynomial keeps them clean K adjoin alpha, smallest field Quotient by minimal gives the yield [Chorus] Field extensions, towers climb Algebraic or transcendental time Simple extensions, minimal degree Tower law multiplies what we see Finite fields with prime power size Perfect laboratories in disguise [Bridge] Tower law declares the truth K to M to L forsooth Degrees multiply along the chain Constraints that keep extensions tamed F sub q has cyclic stars Minus one elements, no bars [Verse 3] Irreducible polynomials build the frame Finite fields play the coding game Every prime power finds its match Unique up to isomorphic catch Galois fields in crypto's heart Where theory meets the practical art [Chorus] Field extensions, towers climb Algebraic or transcendental time Simple extensions, minimal degree Tower law multiplies what we see Finite fields with prime power size Perfect laboratories in disguise [Outro] From rationals to complex plane Field extensions break the chain Basis elements tell the tale Vector spaces never fail
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