A Formalization of Dedekind Domains and Class Groups of Global Fields

Baanen, Anne; Dahmen, Sander R.; Narayanan, Ashvni; Capriglio, Filippo Alberto Edoardo Nuccio Mortarino Majno Di

doi:10.4230/lipics.itp.2021.5

Cited by 5 publications

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“…Due to the distributed organization of mathlib, it is impossible to cite every author who contributed a piece of code that we used. However, we remark that our formalization makes extensive use of the theory of Dedekind domains [4] and of the theory of uniform spaces and completions, originally developed in the perfectoid space formalization project [6].…”

Section: Lean and Mathlibmentioning

confidence: 99%

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Formalizing the Ring of Adèles of a Global Field

Inés¹

2022

Preprint

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The ring of adèles of a global field and its group of units, the group of idèles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we formalized adic valuations on Dedekind domains. We present some applications, including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idèle class group.

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Section: Lean and Mathlibmentioning

confidence: 99%

“…There are several equivalent definitions of Dedekind domain, three of which have been formalized in mathlib [4]. We work with the one formalized in is_dedekind_domain : a Dedekind domain R is an integrally closed Noetherian integral domain with Krull dimension 0 or 1 [14].…”

Section: Dedekind Domains and Adic Valuationsmentioning

confidence: 99%

Formalizing the Ring of Adèles of a Global Field

Inés¹

2022

Preprint

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“…This paper is an extended version of a paper published in the ITP 2021 conference proceedings [3]. The additions to this paper, apart from several clarifications and enhancements throughout the text, mainly concern the following.…”

Section: Introductionmentioning

confidence: 99%

A Formalization of Dedekind Domains and Class Groups of Global Fields

Baanen

Dahmen

Narayanan³

et al. 2022

J Autom Reasoning

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Dedekind domains and their class groups are notions in commutative algebra that are essential in algebraic number theory. We formalized these structures and several fundamental properties, including number-theoretic finiteness results for class groups, in the Lean prover as part of the mathematical library. This paper describes the formalization process, noting the idioms we found useful in our development and ’s decentralized collaboration processes involved in this project.

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“…I assume some familiarity with basic ring and field theory, corresponding to a first course in the subject, and found in many undergraduate texts such as [48,76]. This chapter contains material adapted from the paper "A formalization of Dedekind domains and class groups of global fields", specifically the extended version published in the Journal of Automated Reasoning [11] of a paper presented at the ITP 2021 conference [10], and from the paper "Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves", presented at CPP 2023 [9]. All authors contributed to these papers.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…This chapter describes the formalization process, noting the idioms we found useful in our development and Mathlib's decentralized collaboration processes involved in this project. This chapter is adapted from the paper "A formalization of Dedekind domains and class groups of global fields", specifically the extended version published in the Journal of Automated Reasoning [11] of a paper presented at the ITP 2021 conference [10]. All authors contributed to the formalization project as well as to both versions of the original paper.…”

Section: A Formalization Of Dedekind Domains and Class Groups Of Glob...mentioning

confidence: 99%

Formalizing Fundamental Algebraic Number Theory

Baanen

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The research described in my thesis involves the extension and maintenance of the library of formal mathematics Mathlib for the interactive theorem prover Lean. The process of mechanization that results in formal mathematics rests on a detailed, uniform and unambiguous translation of mathematical knowledge. Thus, it requires suitable design choices, which needs a thorough understanding of the functionality provided by Lean. After giving a general overview of Lean and Mathlib, the thesis extensively covers a specific mechanism of Lean: the instance parameter. This makes possible the design pattern of typeclasses. Related mechanisms for organizing mathematical structures can be found in many other interactive theorem provers. Based on characteristic practical examples, I describe a variety of design choices that help in applying this functionality. Based on this research, changes and additions were made to Mathlib, also on fundamental levels. Many of the design choices are also applied in the subsequent chapters. After this follows a part of the thesis with the emphasis on formal mathematics, beginning with an overview of the mathematics that is covered. As part of a dynamically changing group of contributors, I worked on formalizing subjects that lie at the basis of algebraic number theory, and these formalizations have since been added to Mathlib. An important theme of this chapter is that it becomes possible to treat different occurences of the same situation in an equal manner, if the right formulation is chosen. Next, we apply the previous work to the solution of concrete problems, namely the formal computation of class numbers and a formal description of all integer solutions to certain Diophantine equations. This required us to face the challenge of writing down definitions that are useful both in the case of high abstraction, and when very specific objects are involved. Computing the desired properties for now remains a matter of manually applying calculation steps, although computer algebra systems can supply the outcome (without a formal proof) within mere moments. Finally, I describe proof tactics I developed, that make it possible to automatically prove certain classes of equalities in the study of rings. By applying these, many manual and tedious calculation steps can be left to the computer. The outcome of this research was not merely to extend the list of theorems available in Mathlib, but to enrich the possibilities of effectively formalizing mathematics.

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A Formalization of Dedekind Domains and Class Groups of Global Fields

Cited by 5 publications

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Formalizing the Ring of Adèles of a Global Field

Formalizing the Ring of Adèles of a Global Field

A Formalization of Dedekind Domains and Class Groups of Global Fields

Formalizing Fundamental Algebraic Number Theory

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