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Number bases in quadratic fields. (English) Zbl 0562.12004

An algebraic integer \(\theta\) in a number field K is called a base if every algebraic integer \(\alpha\) in the field K has a representation in the form (*) \(\alpha =r_ 0+r_ 1 \theta +...+r_{\ell} \theta^{\ell}\), where the \(r_ 1\) are rational integers satisfying \(0\leq r_ 1<| N(\theta)|\). The size of the exponent \(\ell\) in (*) is investigated as a function of \(\alpha\) and \(\theta\). For example, in imaginary quadratic fields we show that \(\ell =[\frac{\log N(\alpha)}{\log N(\theta)}]+O(1)\), where O(1) is a constant depending only on K. A similar but more complicated estimate is obtained for real quadratic fields.

MSC:

11R11 Quadratic extensions
11R04 Algebraic numbers; rings of algebraic integers
11A63 Radix representation; digital problems
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