The ring of finite algebraic numbers and its application to the law of decomposition of primes

TL;DR

This paper introduces a method to express finite algebraic numbers using linear recurrent sequences and applies it to characterize prime splitting in finite Galois extensions over rationals.

Contribution

It provides an explicit technique for representing algebraic numbers and links this to prime decomposition in Galois extensions, advancing understanding of algebraic number theory.

Findings

Explicit expression of algebraic numbers via linear recurrent sequences

Characterization of splitting primes in Galois extensions

Enhanced tools for algebraic number analysis

Abstract

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.