About Grothendieck-Witt group of $p$-adic Fields Let $p$ a odd prime number, is well-known that the Square Classes of a rational $p$-adic $\mathbb{Q}_p$ is a group with 4 elements, i.e.
$$
\mathbb{Q}_p^{\times}/\mathbb{Q}_p^{\times 2}=\{1,p,\zeta,p\zeta\}
$$
Where $\zeta$ is a ($p$-1)-th root of unity.
I want to know if exist any reference of the Grothendieck-Witt group $GW(\mathbb{Q}_p)$ and its fundamental ideal $I_{\mathbb{Q}_p}$
Concretly, I want to calculate the group ($\mu_2$ is the group $\{\pm 1\}$)
$$
I_{\mathbb{Q}_p}\otimes\mu_2
$$
and verify if satisfies some properties.

The Grothendieck-Witt and Witt groups of $p$-adic fields $F$ ($p$ odd as in your question) are well-known:
If the cardinality of the residue field $q$ is congruent to $1$ mod $4$, then
$$\text{GW}(F) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}^{\oplus3} \text{ as abelian groups and } \text{W}(F) \cong \mathbb{Z}/2\mathbb{Z}[V_4] \text{ as rings},$$
where $V_4$ is the Klein $4$-group and if $q$ is congruent to $3$ mod $4$, then
$$\text{GW}(F) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}\text{ as abelian groups and } \text{W}(F) \cong \mathbb{Z}/4\mathbb{Z}[C_2] \text{ as rings}.$$
A reference for that is Thm 2.2 of chapter VI in Lam's Introduction to quadratic forms over fields. If you check the details you should probably also be able to see how the fundamental ideals look like.
