The Complexity of Computing all Subfields of an Algebraic Number Field

TL;DR

This paper introduces a method to efficiently compute all subfields of a finite separable algebraic number field by intersecting principal subfields, significantly improving computational speed and complexity when many subfields exist.

Contribution

The paper presents a novel approach for rapidly computing intersections of principal subfields, enhancing the efficiency of finding all subfields in algebraic number fields.

Findings

Faster computation of subfields in algebraic number fields

Reduced complexity in subfield enumeration

Applicable to fields with many subfields

Abstract

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity.