Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x\_1, \ldots , x\_n$ be a $\mathbb{Q}$-basis of $K$ and let $f: K \to \mathbb{Q}$ be a $\mathbb{Q}$-linear map such that $f(x\_i \cdot x\_j)=\delta\_{ij}$ where $\delta\_{ij}$ is the Kronecker delta.


Is it true that the $\mathbb{Z}$-module generated by $x\_1, \ldots, x\_n$ is closed under multiplication by elements of $S$?



lattice in number field already a fractional ideal?
When $\alpha$ is fixed at the beginning, this is not always true, and it doesn't matter whether $K$ is or isn't totally real.


Note first that any full-rank lattice in any number field admits *some* order of the field as endomorphisms. (Proof: The action of any algebraic integer $y$ generating the field on the lattice $\sum x\_i \mathbb{Z}$ by multiplication takes each $x\_j$ to a rational linear combination of the $x\_i$. Let $N$ be the common denominator of the $n^2$ rational coefficients. Then $Ny$ takes each $x\_j$ to an integer linear combination of the $x\_i$, and thus all of $\mathbb{Z}[Ny]$ takes the lattice into itself. Of course, a larger order might still work, too.)


So if you want to conclude that a particular given $\mathbb{Z}[\alpha]$ works, you'll need to impose a condition on the lattice which explicitly depends on $\alpha$, not merely on the multiplication in $K$.


The condition $f(x\_i \cdot x\_j) = \delta\_{ij}$ does admit nontrivial examples. I found one by looking at quadratic fields and trying $x\_1=1$ and algebraic integer $x\_2$; then $f(1\cdot x\_2)=0$ and $f(1^2)=f(x\_2^2)=1$ force $x\_2^2-ax\_2-1=0$ for some $a$, and $a=8$ produces (the unit ideal of) the non-maximal order $\mathbb{Z}[\sqrt{17}]$. This is not a fractional ideal of the maximal order: Multiplication by the algebraic integer $\alpha=(1+\sqrt{17})/2$ does not preserve the lattice. Had we started from $\alpha=\sqrt{17}$ instead, the same lattice and the same $f$ would have worked.


In higher degrees, a power base $(x\_1,x\_2,x\_3,\ldots) = (1,x,x^2,\ldots)$ for the lattice never admits such an $f$ when $n\ge3$, because we'd get $0 = f(1\cdot x\_3) = f(1\cdot x^2) = f(x\_2\cdot x\_2) = 1$, but this obstacle can be avoided by passing to another base of the same lattice. When I start from $x\_1=1$ and algebraic integer $x\_2$, I always seem to end up with one relation among the coefficients of the defining polynomial of $x\_2$. This puts a mild constraint on $K$, but not much of an additional constraint on the lattice. And it implies nothing about the relationship between the lattice and a given $\alpha$.


