Let $F$ be a local field and $F^{\times}$ contains all units. Then what is the structure of $F^{\times}/(F^{\times})^2$ ?
For example if $F=\mathbb{Q}_p$, the p-adic field , then what does $\mathbb{Q}_p^{\times}/(\mathbb{Q}_p^{\times})^2$ means and what is its structure ? How does it help us?