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Sums of integral squares in cyclotomic fields. (English) Zbl 1190.11026

Summary: Let \(K_n=\mathbb Q(\zeta_n)\) be the \(n\)th cyclotomic field with \(n\not\equiv 2\pmod 4\). Let \(O_n=\mathbb Z(\zeta_n)\) be the ring of integers of \(K_n\) and \(S_n\) the set of all elements \(\alpha\in O_n\) which are sums of squares in \(O_n\). Let \(g_n\) be the smallest positive integer \(m\) such that every element in \(S_n\) is a sum of \(m\) squares in \(O_n\). In this Note, we show that \(g_n=3\) unless \(n\) is odd and the order of 2 in \((\mathbb Z/n\mathbb Z)^*\) is odd, in which case \(g_n=4\).

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
11R18 Cyclotomic extensions
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References:

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