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

## TL;DR

This paper determines the bases of the 2-adic logarithm's image on principal units across all quadratic extensions of Q_2, enhancing understanding of its algebraic structure and potential applications.

## Contribution

It provides explicit bases for the 2-adic logarithm's image in all quadratic extensions of Q_2, revealing the module structure of the logarithm.

## Key findings

- Explicit bases for the 2-adic logarithm in quadratic extensions
- Insights into the free module structure of the logarithm
- Applications discussed for algebraic and number theoretic contexts

## 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.