Let $E/F$ be a finite field extension of number field $F$.
$\textbf{Q:}$Can one complete integral basis of $F/Q$ to integral basis of $E/Q$? I doubt this works. Maybe one wants $E$ as a composite field of $F$ with another field $F'$ s.t. they have coprime discriminant. In general, I doubt it is true.
The reason to ask this question is that $V_1\subset V$ is a $R-$vector space. $Q$ is basically taking analogy of the following statement.
$\textbf{Q'1.}$ If I have a lattice $L\subset V$ and consider $L$'s image under $\frac{V}{V_1}$, does the image remain to be a lattice? I encountered the statement for $L$ lattice admitting Riemann form $E$(in other words, torus $\frac{V}{L}$ admits a non-trivial meromorphic section.) and this very riemann form gives the sense of discreteness of the image of lattice under the quotient.(i.e. Image of $L$ is a discrete lattice in the quotient.)
$\textbf{Q'2.}$ Is this lattice discrete without notion of riemann form? It seems that I want to drop the condition of existence of Riemann form here.