what is $\mathbb{Q}_p^\times/\mathbb{Q}_p^{\times^2}$ called?

Question

I have to do a presentation about $p$-adic numbers and I don't know what to call $\mathbb{Q}_p^\times/\mathbb{Q}_p^{\times^2}$. Can you help me, please?

Thank you!

Answer 1

For an arbitrary field $K$, a name used for $K^\times/(K^\times)^2$ is the group of "square classes" (understood not to include $0$). It is a bit more specific than the more general term "quotient group". For example, $\mathbf R$ has two square classes while $\mathbf Q_p$ has four square classes when $p \not= 2$ and eight square classes when $p = 2$. The field $\mathbf Q$ has infinitely many square classes.