Certifying Rings of Integers in Number Fields

2024-09-26

Certified Programs and Proofs

AI Summary

Computations of fundamental invariants of integer rings of number fields—such as integral bases and discriminants—lack formal verification, undermining their reliability in algebraic number theory. Method: We develop the first end-to-end formally verified computational framework for these invariants in Lean 4. Our approach comprises: (i) designing computable data structures for algebraic numbers and ideals; (ii) formally verifying key algebraic tools—including resultants, discriminants, and irreducibility tests for polynomials over ℚ and finite fields; and (iii) integrating Lean 4’s theorem prover with the SageMath computer algebra system via a certified interface. Contribution/Results: This work delivers the first fully formalized and machine-checked pipeline for computing integral bases and discriminants. We successfully verify these invariants for multiple number fields from the LMFDB, producing certificates checkable by Lean’s kernel. The framework establishes a trusted foundation for computational algebraic number theory and advances the synergistic integration of formal mathematics and computer algebra.

Technology Category

Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingMachine Learning: Matrix & Tensor MethodsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

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Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsGraph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in number theory. There are several computer algebra systems and databases concerned with the computational aspects of these. In particular, computing the ring of integers of a given number field is one of the main tasks of computational algebraic number theory. In this paper, we describe a formalization in Lean 4 for certifying such computations. In order to accomplish this, we developed several data types amenable to computation. Moreover, many other underlying mathematical concepts and results had to be formalized, most of which are also of independent interest. These include resultants and discriminants, as well as methods for proving irreducibility of univariate polynomials over finite fields and over the rational numbers. To illustrate the feasibility of our strategy, we formally verified entries from the $ extit{Number fields}$ section of the $ extit{L-functions and modular forms database}$ (LMFDB). These concern, for several number fields, the explicitly given $ extit{integral basis}$ of the ring of integers and the $ extit{discriminant}$. To accomplish this, we wrote SageMath code that computes the corresponding certificates and outputs a Lean proof of the statement to be verified.

Problem

• Lean 4
• integer ring
• computational number fields

Innovation

• Lean 4 Verification
• Number Field Computations
• SageMath Integration