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Suppose $n ≥ 2$ , and $f(x)=a_n x^n+a_{n-1} x^{n-1}+⋯+a_1 x+a_0∈\mathbb Z[X]$ with $a_n≠0$.

$M_{1}, M_{2},...,M_{r}$ are the distinct roots of $f(x)$ over $\mathbb C$.

Show that $a_{n}M_{1}M_{2}...M_{r}∈\mathbb Z$.

I know that $a_{n}M_{1}$ is an algebraic integer.

NOTE that the roots we have just the distinct ones and so we can not use the fundamental theorem of algebra.

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ks1
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    What tools are you aware of that might help in solving this problem? Are there any attempts you have made so far? – abiessu Sep 22 '20 at 01:31
  • Hint: you can find an exact formula for $a_nM_1M_2\cdots M_r$ in terms of the other coefficients of the polynomial. – Greg Martin Sep 22 '20 at 01:43
  • The product is an algebraic integer, if we prove that the product is also rational number we'll get the result – ks1 Sep 22 '20 at 01:44
  • @ks1: You are overthinking this. This is an elementary problem. There is no need to use anything beyond high school algebra. – tomasz Sep 22 '20 at 01:46
  • note that we have r roots not n – ks1 Sep 22 '20 at 01:48
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    Hint: the radical $,{r}(f) := f/\gcd(f,f')\in \Bbb Q[x],$ has no repeated roots, and can be scaled by a rational to be a factor of $f$ in $\Bbb Z[x],$ by Gauss's Lemma. – Bill Dubuque Sep 22 '20 at 09:15

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