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Hilbert’s Error?

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Notes

  1. A similar alternative appears when we prove that the general fifth-degree equation cannot be solved in radicals: one could prove that a general formula does not exist, or give an example of an equation with integer coefficients whose roots cannot be obtained from rational numbers by arithmetic operations and taking roots. These two settings are distinct.

  2. For such a construction, see the Wikipedia article on the incircle and excircles of a triangle.

  3. We consider only constructions with straightedge; the initial configuration may contain circles, but new circles cannot be added. In allowing the use of a compass, we should add the corresponding operations in both the positive and negative definitions; they remain equivalent.

  4. This proof essentially follows [1], though the authors do not give a positive version of the definition explicitly, speaking about “algorithms” instead.

  5. We can avoid the cardinality argument by using algebraic objects and a transformation in our family that maps the center to a nonalgebraic point, as is done in [1].

References

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Acknowledgments

I am grateful to Arseny Akopyan and Roman Fedorov for sharing their paper [1] and its preliminary versions, and for discussing their results; to Sergey Markelov, who participated in these discussions and had many interesting suggestions; to Sergey Lvovsky for critical remarks; to Rupert Hölzl, who corrected the translation of German quotations and made many useful remarks. The author was supported by RaCAF ANR-15-CE40-0016-01 and RBFR 16-01-00362 grants while working on this paper.

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  1. LIRMM, CNRS, and University of Montpellier (On leave from IITP RAS), 161 rue Ada, Montpellier, 34095, France

    Alexander Shen

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  1. Alexander Shen
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Correspondence to Alexander Shen.

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For Rafail Gordin, on his 70th birthday.

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Shen, A. Hilbert’s Error?. Math Intelligencer 40, 6–11 (2018). https://doi.org/10.1007/s00283-018-9792-8

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