We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{ Q}_2$. We prove that if $K$ is one of $\{ \mathbb{ Q}_2(\sqrt{ 2}), \mathbb{ Q}_2(\sqrt{ 10}), \mathbb{ Q}_2(\sqrt{ -2}), \mathbb{ Q}_2(\sqrt{ -10})\}$, $\Gamma^*(d,K) = \frac{ 3}{ 2}d$, and if $K$ is one of $\{ \mathbb{ Q}_2(\sqrt{ -1}), \mathbb{ Q}_2(\sqrt{ -5})\}$, $\Gamma^*(d,K) = d+1$. The case $d=6$ was previously known.
Categories: math.NT