On the bases of the image of $2$-adic logarithm on the group of principal units

Abstract

This paper computes the bases of the image of $2$-adic logarithm on the group of the principal units in all 7 quadratic extensions of ${\mathbb{Q}}_2$. This helps one to understand the free module structure of $2$-adic logarithm at arbitrary points on its domain. We discuss some applications at the end.

arXiv comment18 pages, we have revised the previous versions, included applications of our results, slight change in title, and added more references. Comments are welcome