Solubility of Additive Quartic Forms over Ramified Quadratic Extensions of $\mathbb{Q}_2$

We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})$ of $\mathbb{Q}_2$. In all four fields, we prove that $\Gamma^*(4,K) = 11$. This is the first example of such a computation for a proper extension of $\mathbb{Q}_p$ where the degree is a power of $p$ greater than $p$.