Let \( \alpha \) be an algebraic number of degree \( d\geq 2 \). We consider the set \( E(\alpha) \) of positive integers \( n \) such that the primitive \( n \)th root of unity \( e^{2\pi i/n} \) is expressible as a quotient of two conjugates of \( \alpha \) over \( {𝕈} \). In particular, our results imply that \( E(\alpha) \) is small. We prove that \( |E(\alpha)|<d^{\frac{c}{\log\log d}} \), where \( c=1.04 \) for each sufficiently large \( d \). We also show that, in terms of \( d \), this estimate is best possible up to a constant, since the constant \( 1.04 \) cannot be replaced by any number smaller than \( 0.69 \).

让\( \alpha \)是一个度数\( d\geq 2 \)的代数数。我们考虑正整数 集合\( E(\alpha) \) \( n \) 使得基本的\( n \) th 单位根\( e^{2\pi i/n} \) 是可表达的作为\( \alpha \)在 \( {𝕈} \) 上的两个共轭的商 。特别是，我们的结果意味着 \( E(\alpha) \)很小。我们证明 \( |E(\alpha)|<d^{\frac{c}{\log\log d}} \)，其中\( c=1.04 \)对于每个足够大的 \( d \)。我们还表明，就 \( d \)，这个估计最有可能达到一个常数，因为常数 \( 1.04 \)不能被任何小于\( 0.69 \) 的数字替换。