One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,\*, U, \mid )$. In this Language $U$ is the unary predicate standing for the Valuation Ring of the model, and $\mid $ is a binary relation such that $x\mid y \leftrightarrow \exists z\in U \ x\*z=y$. How do you prove the completeness of this theory in that language?



Completeness of Algebraically Closed Valued Fields(ACVF) Theory
Just like with algebraically closed fields, for completeness you need to specify the characteristic of the field and the characteristic of the residue field. 


There is a general trick if you have a theory $T$ with quantifier elimination and a structure $A$ that is embedded in every model of $T$, then $T$ is complete. Let $M$ and $N$ be models of $T$ and let $\phi$ be any sentence. There is a quantifier free sentence $\psi$
that is equivalent to $\phi$ in models of $T$. But, since $\psi$ is quantifier free,
$$M\models \psi\Leftrightarrow A\models\psi\Leftrightarrow N \models\psi.$$ Thus
$M\models\phi\Leftrightarrow N\models \phi$.


Now, suppose you are looking at characteristic 0 fields with characteristic p residue field. The rationals with the p-adic valuation are a substructure of any model of ACVF of characteristic $0$ with characteristic $p$ residue field. 


