Algebraically Closed Fields in Isabelle/HOL

Abstract

A fundamental theorem states that every field admits an algebraically closed extension. Despite its central importance, this theorem has never before been formalised in a proof assistant. We fill this gap by documenting its formalisation in Isabelle/HOL, describing the difficulties that impeded this development and their solutions.

Developing Field Theory in Mizar

“…Lean provides the theory up to the fundamental theorem of Galois theory [5]. General field theory also has been formalized in Isabelle -in particular the existence of algebraic closures of fields has been proved [8].…”

Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types

“…And while these formalizations constitute an impressive body of work, parts of it are deprecated code not using locales or the Isar language, an additional layer of vernacular which allows for structured proofs making them more legible and easier to maintain. Despite these problems, there are noteworthy achievements like the formalisation of the algebraic closure of a field [13].…”

Formalizing Galois Theory

“…More generally, results in field theory have been formalized in several other proof assistants, including Isabelle, Mizar, and HOL Light. The development of field theory in Isabelle is especially advanced-for example, the existence of algebraic closures of fields was recently proved in Isabelle by Paulo Emílio de Vilhena and Lawrence Paulson [7]. Also, the impossibility of trisecting an angle with a ruler and compass has been proved in both Isabelle and HOL Light (though notably the proof in HOL Light avoids talking about field extensions) and a number of results about fields have been formalized in Mizar, including the construction of a field extension adjoining a root of a given polynomial [12].…”