Principal Well-Rounded Ideals of real quadratic fields

TL;DR

This paper establishes necessary and sufficient conditions for real quadratic fields to possess principal well-rounded ideals, proves their infinitude, and introduces efficient algorithms for constructing such ideals, including prime ones.

Contribution

It combines classical Pell equation results with Srinivasan's conditions to characterize principal well-rounded ideals and develops fast algorithms for their construction.

Findings

Infinitely many real quadratic fields have principal well-rounded ideals.

Constructed algorithms are efficient and practical for large discriminants.

Existence of infinitely many fields with prime PWR ideals is shown.

Abstract

It has been well known since Gauss that the principality of an ideal in a real quadratic field $K$ is equivalent to the solvability of a certain generalized Pell equations. In this paper, we combine this classical result with Srinivasan's conditions for the existence of well-rounded ideals in $K$ to obtain necessary and sufficient criteria for a real quadratic field to have principal well-rounded (PWR) ideals. Using these criteria, we prove that there are infinitely many real quadratic fields that have PWR ideals. Moreover, these ideals are pairwise non-similar. We then construct new algorithms that produce these PWR ideals, especially when the field discriminant is large. Our algorithms run in sub-exponential time theoretically; however, they are very fast in practice by employing some commonly used probabilistic algorithms for testing squarefreeness. Finally, we briefly consider criteria for the existence of prime PWR ideals and show that there are infinitely many real quadratic fields that have prime PWR ideals.