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Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a $\mathbb{Z}$-basis for it. For an explicit number field one can find thus a $\mathbb{Z}$-basis for it's ring of integers by explicit calculation. Suppose you start the other way around, that you have some set of elements $\{x_i\}_{i\in \{1,2,\ldots,n\}}$ of $K$ for which you think it's a $\mathbb{Z}$-basis for $\mathcal{O}_K$. How do you proceed then?

I know that their is a lot of linear algebra for number fields, so probably that can be put to use. But it makes often use of embeddings of $K$ into $\mathbb{C}$ (or $\mathbb{R}$), which can be tricky to work with. I am interested in general theorems for proving that some set is a $\mathbb{Z}$-basis, but any help is appreciated.

user299843
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    This is clear for quadratic number fields, but already difficult for cubic number fields, e.g., see here, here, or here. – Dietrich Burde Feb 12 '16 at 11:19
  • Indeed, without further information it's really difficult to determine if something is a $\mathbb{Z}$-basis. I was hoping that with some extra information, such as knowing all the embeddings, that there are some theorems for proving it. – user299843 Feb 12 '16 at 12:03
  • In the book "number fields" of Marcus are some results, e.g., using ramification index etc.. For example, see here. – Dietrich Burde Feb 12 '16 at 16:30

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