On the indices of multiquadratic number fields (Q2883258)

Let \(K\) be an algebraic number field with ring of integers \(\mathbb Z_K\). The index of an algebraic integer \(\alpha\) (generating \(K\) over \(\mathbb Q\)) is defined by the index of the additive group of \(\mathbb Z[\alpha]\) in the additive group of \(\mathbb Z_K\), that is \(I(\alpha)=(\mathbb Z_K^+:\mathbb Z[\alpha]^+).\) The field index of \(K\) is the gcd of all relevant indices of algebraic integers of \(K\).NEWLINENEWLINEConsider multiquadratic number fields of type \(K=\mathbb Q(\sqrt{a_1},\ldots,\sqrt{a_r})\). In case \(r=2\) \textit{I. Gaál, A. Pethő} and \textit{M. Pohst} [Arch. Math. 57, No. 4, 357--361 (1991; Zbl 0724.11049)] gave an explicit formula for the field index of biquadratic fields. For \(r\geq 2\), \textit{B. Schmal} [Arch. Math. 52, No. 3, 245--257 (1989; Zbl 0684.12006)] gave explicitly an integral basis of \(K\). This allows the authors of the present paper to consider the field index of any multiquadratic number field. They use a very accurate factorization of the index form corresponding to the integral basis of \(K\) and omit any calculation by computer. They describe the index of \(K\) completely for \(r=3\). They prove that for any prime power \(p^k\), the index of \(K\) is divisible by \(p^k\) if \(r\) is large enough.