**This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.**

### Abstract

Based on existing libraries for matrices, factorization of rational polynomials, and Sturm's theorem, we formalized algebraic numbers in Isabelle/HOL. Our development serves as an implementation for real and complex numbers, and it admits to compute roots and completely factorize real and complex polynomials, provided that all coefficients are rational numbers. Moreover, we provide two implementations to display algebraic numbers, an injective and expensive one, or a faster but approximative version.

To this end, we mechanized several results on resultants, which also required us to prove that polynomials over a unique factorization domain form again a unique factorization domain.

### License

### History

- April 16, 2017
- Use certified Berlekamp-Zassenhaus factorization, use subresultant algorithm for computing resultants, improved bisection algorithm
- January 29, 2016
- Split off Polynomial Interpolation and Polynomial Factorization

### Topics

### Session Algebraic_Numbers

- Algebraic_Numbers_Prelim
- Bivariate_Polynomials
- Resultant
- Algebraic_Numbers
- Sturm_Rat
- Factors_of_Int_Poly
- Min_Int_Poly
- Algebraic_Numbers_Pre_Impl
- Cauchy_Root_Bound
- Real_Algebraic_Numbers
- Real_Roots
- Complex_Roots_Real_Poly
- Compare_Complex
- Interval_Arithmetic
- Complex_Algebraic_Numbers
- Show_Real_Alg
- Show_Real_Approx
- Show_Real_Precise
- Algebraic_Number_Tests
- Algebraic_Numbers_External_Code

### Used by

- A First Complete Algorithm for Real Quantifier Elimination in Isabelle/HOL
- Factorization of Polynomials with Algebraic Coefficients
- Solving Cubic and Quartic Equations
- The BKR Decision Procedure for Univariate Real Arithmetic
- The Hermite–Lindemann–Weierstraß Transcendence Theorem
- A verified LLL algorithm
- Linear Recurrences