It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented.
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Supported by RFBR (project No. 17-01-00247) and by the RF Ministry of Science and Higher Education (state assignment to Sobolev Institute of Mathematics, SB RAS, project No. 0314-2019-0002).
The work was carried out at the expense of the subsidy allocated to Kazan (Volga Region) Federal University for the fulfillment of the state assignment in the sphere of scientific activity, project No. 1.13556.2019/13.1.
Translated from Algebra i Logika, Vol. 58, No. 6, pp. 673-705, November-December, 2019.
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Alaev, P.E., Selivanov, V.L. Fields of Algebraic Numbers Computable in Polynomial Time. I. Algebra Logic 58, 447–469 (2020). https://doi.org/10.1007/s10469-020-09565-0
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DOI: https://doi.org/10.1007/s10469-020-09565-0