Grossman, Edward H. Number bases in quadratic fields. (English) Zbl 0562.12004 Stud. Sci. Math. Hung. 20, 55-58 (1985). 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. Cited in 2 Documents MSC: 11R11 Quadratic extensions 11R04 Algebraic numbers; rings of algebraic integers 11A63 Radix representation; digital problems Keywords:number bases; algebraic integer; quadratic fields PDFBibTeX XMLCite \textit{E. H. Grossman}, Stud. Sci. Math. Hung. 20, 55--58 (1985; Zbl 0562.12004)