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Universal quadratic forms over multiquadratic fields. (English) Zbl 1428.11071

A totally positive definite quadratic form with coefficients in the ring of integers \(\mathcal O_K\) of a totally real number field \(K\) is said to be universal if it represents all totally positive elements of \(\mathcal O_K\). Denote by \(m(K)\) the minimal rank of such a universal form.
V. Blomer and the first author [Math. Proc. Camb. Philos. Soc. 159, No. 2, 239–252 (2015; Zbl 1371.11084); Bull. Aust. Math. Soc. 94, No. 1, 7–14 (2016; Zbl 1345.11025)] used continued fractions to construct, for each \(N\), infinite families of real quadratic fields \(K\) such that \(m(K)\geq N\).
In the present paper this result is extended to: For all pairs of positive integers \(k,N\), there are infinitely many totally real multiquadratic fields \(K\) of degree \(2^k\) over \(\mathbb Q\) such that \(m(K)\geq N\). The general approach is to produce sufficiently many additively indecomposable integers satisfying suitable additional conditions. The argument proceeds by induction on \(k\), starting with a quadratic subfield of \(K\).

MSC:

11E12 Quadratic forms over global rings and fields
11R20 Other abelian and metabelian extensions
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References:

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