### Algebraic Numbers in Isabelle/HOL

René Thiemann and Akihisa YamadaProceedings of the 7th International Conference on Interactive Theorem Proving (ITP 2016), Lecture Notes in Computer Science 9807, pp. 391 – 408, 2016.

### Abstract

We formalize algebraic numbers in Isabelle/HOL, based on existing libraries

for matrices and Sturm’ s theorem. Our development serves as a verified

implementation for real and complex numbers, and it admits to compute roots

and completely factor 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, and a faster

but approximative version.

To this end, we mechanize several results on resultants, which also required

us to prove that polynomials over a unique factorization domain form again

a unique factorization domain. We moreover formalize algorithms for

factorization of integer polynomials: Newton interpolation, factorization

over the integers, and Kronecker’ s factorization algorithm, as well as a

factorization oracle via Berlekamp’ s algorithm with the Hensel lifting.

### BibTeX

@inproceedings{RTAY2016-ITP, author = "Ren{\'e} Thiemann and Akihisa Yamada", title = "Algebraic Numbers in Isabelle/HOL", booktitle = "Proceedings of the 7th International Conference on Interactive Theorem Proving", editor = "Jasmin Christian Blanchette and Stephan Merz", series = "Lecture Notes in Computer Science", volume = 9807, pages = "391--408", year = 2016, doi="10.1007/978-3-319-43144-4_24" }