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The integral trace form as a complete invariant for real $S_n$ number fields

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arxiv 1812.03133 v6 pith:5HXJWFVP submitted 2018-12-07 math.NT

classification math.NT
keywords fieldsintegralnumbertracerealcharacterizecompletefield
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abstract

In the past the first named author has studied to what extent the integral trace can characterize a number field beyond what the discriminant does. The cases of cyclic number fields and non-totally real fields are more or less settled, concluding that for such fields the integral trace does not always characterize the field. In this paper we show that the integral trace is a complete invariant for degree $n$, $S_n$ real number number fields that satisfy certain ramification bound. Among the real $S_n$ fields that our results cover, there are those of square free different ideal. Moreover, for such fields we find an explicit description of the isometry group of the integral trace.

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Cited by 1 Pith paper

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  1. An introduction to $\Gamma$-number fields

    math.NT 2019-08 reject novelty 6.0 of 10

    For Gamma-number fields, the spinor genus of the integral trace form is determined exactly by the discriminant and the signature.

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