Un dénominateur commun pour les racines d'un polynôme scindé

Un dénominateur commun pour les racines d’un polynôme scindé

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Soit $P \in \mathbb{Z}[X]$ unitaire^EN et $\gamma \in \mathbb{Q}$ . Montrer que si $P+\gamma$ est scindé sur $\mathbb{Q}$ , alors toutes ses racines ont le même dénominateur sous forme irréductible.

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Solution by visitorEN

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Let $a/q$ (lowest terms) be one root and put $g(Y)=q^{\deg f}f(Y/q)$ , a monic polynomial with integer coefficients. Then $q^{\deg f}\gamma=-g(a)$ is an integer, so $g+q^{\deg f}\gamma$ is monic in $\Z[Y]$ , and its rational roots, which are $q$ times the roots of $f+\gamma$ , are integers. So every root has denominator dividing $q$ ; running the argument from each root, the denominators all coincide.

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