Counting principal ideals of small norm in the simplest cubic fields

2025/01/12 by Zindulka, Mikuláš
#11R16 #11R80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.06889

Abstract

We estimate the number of principal ideals I of norm N(I) ≤ x in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary x ≥ 1 , even when x is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.

Related

• Counting ideals in abelian number fields
• The least prime ideal in a given ideal class
• Reduced Ideals in Pure Cubic Fields
• Small ideals in polynomial rings and applications
• Principal Well-Rounded Ideals of real quadratic fields