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    "plainLanguage": "Take a polynomial whose coefficients and inputs are complex numbers. If its degree is positive, then at least one complex number makes the polynomial equal to zero.",
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      "alt": "An ivory theorem poster moves from the positive-degree condition for a complex polynomial to one luminous complex root, with a four-step reading guide and an exact-scope boundary.",
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      "caption": "Positive degree leads to the existence of at least one complex root; the poster visibly limits the claim to root existence.",
      "description": "The poster places deg f > 0 on the left and the existential root formula on the right. Below, evaluation paths on a stylized complex plane converge toward one highlighted zero. A numbered reading guide and red exact-scope panel state that uniqueness, factorization, and multiplicity are not claimed.",
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      "title": "Fundamental Theorem of Algebra at a glance",
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      "transcript": "COMPLEX POLYNOMIALS · ROOT EXISTENCE\nFUNDAMENTAL THEOREM OF ALGEBRA\ndeg f > 0\nThere exists a complex number z\nwith f(z) = 0\nPOSITIVE DEGREE\nCOMPLEX COEFFICIENTS\nAT LEAST ONE ROOT\nHOW TO READ IT\n1 · Choose a complex polynomial f\n2 · Require positive degree\n3 · Find a complex number z\n4 · Evaluate f(z) = 0\nEXACT SCOPE\nRoot existence only. No uniqueness, factorization, or multiplicity count."
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      "alt": "A many-sheeted ivory polynomial surface narrows toward one luminous gold point on an abstract complex plane.",
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      "caption": "A positive-degree complex polynomial is pictured descending to a root where it vanishes.",
      "description": "An engraved ivory surface folds across a dark complex-plane field. Emerald, cobalt, and gold trajectories descend from several sheets toward one bright central point, emphasizing existence of a root without depicting a proof.",
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      "title": "Complex polynomial root schematic",
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      "introduction": "Mathlib proves the exact root-existence theorem by assuming a positive-degree complex polynomial has no root, studying its differentiable reciprocal, using polynomial growth and Liouville's theorem to force that reciprocal to vanish everywhere, and deriving the impossible conclusion that the polynomial is the zero constant.",
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          "alt": "An engraved forest-green polynomial ribbon folds across a circular complex-plane disk while flowing around an empty antique-gold socket.",
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          "caption": "Assume for contradiction that the positive-degree complex polynomial never vanishes, so its evaluation ribbon avoids the root socket everywhere.",
          "description": "The proof begins by negating the desired root existence. Under this contradiction hypothesis, f.eval z is nonzero for every complex z, which supplies the pointwise nonvanishing condition needed to differentiate its reciprocal.",
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          "title": "Assume the positive-degree polynomial has no root",
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          "equation": "0 < f.degree and ∀ z : ℂ, f.eval z ≠ 0",
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          "alt": "On the recurring complex-plane disk, green polynomial waves expand outward while blue reciprocal droplets contract inward toward an empty gold socket.",
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          "caption": "With no zero to obstruct inversion, the reciprocal is differentiable; polynomial growth at infinity becomes reciprocal decay to zero.",
          "description": "Because f.eval is everywhere nonzero, its reciprocal is differentiable. Positive degree makes the norm of the polynomial tend to infinity along the complex cobounded filter, and inversion turns that growth into convergence of the reciprocal to zero.",
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          "title": "Reciprocal decay follows polynomial growth",
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          "equation": "‖f(z)‖ → ∞ and (f.eval z)⁻¹ → 0 as z tends to complex infinity",
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          "alt": "Blue reciprocal droplets distributed across the recurring disk collapse toward uniform tiny gold points over a faint green ribbon field.",
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          "caption": "A differentiable reciprocal that tends to zero at complex infinity is forced by Liouville's theorem to equal zero at every point.",
          "description": "Mathlib applies the Liouville consequence apply_eq_of_tendsto_cocompact to the differentiable reciprocal and its limit at complex infinity. The conclusion is pointwise and global: for every complex z, the reciprocal of f.eval z equals zero.",
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          "title": "Liouville forces the reciprocal to vanish everywhere",
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          "equation": "∀ z : ℂ, (f.eval z)⁻¹ = 0",
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          "caption": "Pointwise zero reciprocals force the polynomial to be the zero constant, contradicting positive degree; therefore a complex root exists.",
          "description": "Inverse injectivity and polynomial extensionality turn the pointwise reciprocal equation into f = C 0. That makes the polynomial's degree incompatible with the original positive-degree hypothesis, so the rootless assumption is false and a complex root exists.",
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          "sha256": "sha256:7cb435ca754b8aa9993bb7a6c812b95963b8bdf0806d1d39c93bd4fd91393ce8",
          "title": "The zero polynomial contradiction reveals a root",
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          "equation": "f = C 0, contradicting 0 < f.degree; hence ∃ z : ℂ, f.IsRoot z",
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      "title": "A Liouville proof of root existence"
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    "whyLandmark": "The theorem says that complex numbers contain the roots needed by every nonconstant one-variable complex polynomial. It is a foundational bridge between algebra and complex analysis."
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  "summary": "Every complex polynomial of positive degree has a complex root.",
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  "title": "Fundamental Theorem of Algebra",
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