This research was supported by the Austrian Science Fund (FWF) project Y757. The authors are listed in alphabetical order regardless of individual contributions or seniority. Sebastiaan is now working at University of Twente, the Netherlands, and Akihisa at National Institute of Informatics, Japan. We thank the reviewers of this paper for their helpful comments.
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We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle’s code generator. The development combines various existing formalizations such as matrices, Sturm’s theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.