关于Craven问题和Belyi定理
On a Question of Craven and a Theorem of Belyi
摘要：
本文证明了一个有理系数多项式能整除某个分裂成$\Q$的多项式的导数当且仅当它的所有无理根都是实单根。这为托马斯·克雷文提出的一个问题提供了答案。类似的想法也导致一个变化的证明贝利定理，每一个代数曲线定义的代数数域承认一个地图$P^1$这是唯一分歧以上三点。事实证明，这种变化以前被G。Belyi本人和Leonardo Zapponi。
英文摘要：
In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\Q$ if and only if all of its irrational roots are real and simple. This provides an answer to a question posed by Thomas Craven. Similar ideas also lead to a variation of the proof of Belyi's theorem that every algebraic curve defined over an algebraic number field admits a map to $P^1$ which is only ramified above three points. As it turned out, this variation was noticed previously by G. Belyi himself and Leonardo Zapponi.
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