7 problems

2-stage Euclideanity conjecture. Every real quadratic field of class number $1$ is 2-stage euclidean.

Let $K$ range over real cubic number fields with class number $1$, and let a field be norm-Euclidean when its ring of integers is Euclidean with respect to the absolute value of th…

A complex cubic field is a cubic number field with nonreal embeddings, and it is norm-Euclidean when its ring of integers is Euclidean with respect to the absolute value of the fie…

Pure cubic inhomogeneous-minimum conjecture.

Inhomogeneous-minimum equality conjecture. For every number field $K$,

Heilbronn's conjecture. There are infinitely many totally real cubic fields that are norm-Euclidean.

For a number field $K$, let … be the Euclidean minimum at $\xi$, and define … The attainment conjecture. The supremum $M(K)$ is a maximum for every number field $K$. This is known…