Algebraic Number Theory and Computational Techniques

Algebraic number theory explores the arithmetic of algebraic numbers, emphasising both theoretical structures and computational strategies that enable explicit determination of key invariants. At its core lies the study of number fields, finite extensions of the rationals, and their rings of integers, which serve as analogues of the ordinary integers. Fundamental invariants such as discriminants, class groups and unit groups capture the arithmetic complexity of these fields. Recent decades have witnessed powerful algorithmic advances: lattice basis reduction techniques facilitate the determination of integral bases and unit computations, while p-adic algorithms and Newton polygon methods enable efficient factorisation of ideals and analysis of ramification. These computational tools underpin applications ranging from explicit class field constructions to cryptographic schemes based on class group arithmetic. Meanwhile, the interplay between theoretical advances—particularly in monogenity and Galois module structure—and practical algorithms continues to broaden both the scope and precision of explicit arithmetic in number fields.

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Technical terms

Number field: A finite extension of the field of rational numbers.

Ring of integers: The integral closure of the integers in a number field, generalising the ordinary integers.

Discriminant: An integer invariant measuring arithmetic complexity and ramification behaviour in a number field.

Monogenic: Describing a number field whose ring of integers is generated by a single element.

Integral basis: A set of algebraic integers that forms a Z-basis for the ring of integers.

Newton polygon: A graphical tool depicting p-adic valuations of polynomial coefficients to analyse factorisation and ramification.

Lattice basis reduction: An algorithmic technique to find short vectors in Euclidean lattices, used to compute integral bases and fundamental units.

References

1. On Indices of Septic Number Fields Defined by Trinomials x7 + ax + b. Mathematics (2023).
2. Newton polygons of higher order in algebraic number theory. Transactions of the American Mathematical Society (2011).
3. Monogenity and Power Integral Bases: Recent Developments. Axioms (2024).
4. Discriminants of fields generated by polynomials of given height. Israel Journal of Mathematics (2023).