Explicit Factorization of Prime Integers in Quartic Number Fields defined by $X^4+aX+b$

TL;DR

This paper provides an explicit method to factor prime ideals in quartic number fields defined by specific quartic polynomials, enhancing understanding of their ideal structure and prime decomposition.

Contribution

It introduces a new explicit factorization technique for prime ideals in quartic fields defined by $X^4+aX+b$, expanding computational tools in algebraic number theory.

Findings

Explicit prime ideal factorizations for all primes in these fields

Enhanced understanding of prime decomposition in quartic fields

Potential applications in computational algebraic number theory

Abstract

For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$.