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If we have two Polynomials $f(x)$ and $g(x)$ with integer coefficients such that $$f(x) \equiv g(x)\left( {\bmod n} \right),n \in {\mathbb{Z}^ + }$$

Does this mean that the coefficients of $f(x)$ are congruent coefficients of $g(x)$ mod n?

If it's True

Why ${x^{p}} \equiv x\left( {\bmod p} \right) \Rightarrow 1 \equiv 0\left( {\bmod p} \right)$?

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up vote 5 down vote accepted

As your example shows, the coefficients are not necessarily congruent. Here is a somewhat different general example. Let $n \ge 2$. Let $P(x)$ be the polynomial $(x-1)(x-2)\cdots (x-n)$, and let $Q(x)$ be the zero polynomial. Then $P(k)\equiv Q(k)\equiv 0\pmod{n}$ for all $k$. But the coefficients certainly do not match modulo $n$.

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This is why it might be better to write $f \equiv g$ instead of $f(X) \equiv g(X)$ if you mean that the coefficients should be congruent. $X^p$ and $X$ are two distinct polynomials in $\mathbb{Z}/p\mathbb{Z}[X]$ which happen to take the same values everywhere.

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There is a good discussion of exactly these two concepts (congruent coefficients versus congruent values) in the first chapter of Borevich and Shafarevich: Number Theory - a good read.

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