Let $p$ be any prime and $k = \mathbb{Q}_p$. The structure of the square class group $k^*/k^{*2}$ is well known. It has four or eight elements depending on whether $p$ is odd or not. If we set
$K = \mathbb{Q}_p(t)$
then $K^*/K^{*2}$, will contain all elements from $k^*/k^{*2}$ multiplied by $1$ and by $t$. But what about sums? For $p \neq q \in k^*/k^{*2}$ will $q+t$ be an additional square class?
In the case of $k = \mathbb{Q}_p$ every element $q$ added to $p$ for example, will be in the same square class mod $p$ obviously, so there is not need to consider sums.