Polynomials, Galois groups, and database-driven arithmetic

This paper presents an ongoing, long-term project in Galois theory, a cornerstone of algebra, by leveraging computational methods to study polynomials and their Galois groups. We introduce large databases of irreducible polynomials as a powerful tool to analyze Galois groups and explore broader arithmetic applications, such as invariants and polynomial heights. By applying computational techniques to these databases, we aim to simplify the determination of solvability by radicals and uncover new patterns in Galois theory. This summary outlines the project’ background, database-driven methodology, potential arithmetic applications, and challenges of integrating computational approaches with classical algebra.