Let $P(x) \in \mathbb{Z}[x]$, $\deg (P(x)) = 4$. Suppose for all $t \in \mathbb{Z}$, $7 \mid P(t)$. Show that all coefficients of $P(x)$ are divisible by $7$.
Should we use the Eisenstein criterion in this problem?
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Hint $ $ If $\,7\nmid f(x)\,$ then $\bmod 7\!:\ 0\not\equiv f$ has more roots than its degree: $\,x\equiv 0,1,2,\ldots, 6$
Bill Dubuque
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Recall that over a field (or domain) a nonzero polynomial has no more roots than its degree, and $,\Bbb F_7 = \Bbb Z/7\Bbb Z,$ is a field. – Bill Dubuque Jun 02 '19 at 04:11