# Implementing field extensions of the form Q[sqrt(b)]

 Title: Implementing field extensions of the form Q[sqrt(b)] Author: René Thiemann (rene /dot/ thiemann /at/ uibk /dot/ ac /dot/ at) Submission date: 2014-02-06 Abstract: We apply data refinement to implement the real numbers, where we support all numbers in the field extension Q[sqrt(b)], i.e., all numbers of the form p + q * sqrt(b) for rational numbers p and q and some fixed natural number b. To this end, we also developed algorithms to precisely compute roots of a rational number, and to perform a factorization of natural numbers which eliminates duplicate prime factors. Our results have been used to certify termination proofs which involve polynomial interpretations over the reals. Change history: [2014-07-11]: Moved NthRoot_Impl to Sqrt-Babylonian. BibTeX: @article{Real_Impl-AFP, author = {René Thiemann}, title = {Implementing field extensions of the form Q[sqrt(b)]}, journal = {Archive of Formal Proofs}, month = feb, year = 2014, note = {\url{https://isa-afp.org/entries/Real_Impl.html}, Formal proof development}, ISSN = {2150-914x}, } License: GNU Lesser General Public License (LGPL) Depends on: Deriving, Show, Sqrt_Babylonian Used by: QR_Decomposition Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.