Let $E/F$ be a quadratic extension of **$p$-adic fields**. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $ \text{id} \neq \sigma \in \text{Gal}(E/F)$.

With respect to this involution, a matrix $X$ can be "hermitian". A way to construct such matrices is by symmetrization: $X=Y \cdot \sigma(Y)^{tr}$.

Given a hermitian $X$, what do we know about the existence of such a $Y$?

If we were talking about $\textbf{C}/\textbf{R}$, the existence of such a $Y$ would be equivalent to $X$ being "positive semidefinite" and the expression $X=Y \cdot \sigma(Y)^{tr}$ would be called the "Cholesky decomposition". But we're not.