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Let's consider this theorem:

Let $f(x)$, $g(x)$, $p(x)\in F[x]$ with $p(x)\neq 0$. We say $f(x)$ is congruent to $g(x)$ modulo $p(x)$ if $p(x)$ divides $f(x)−g(x)$, and we write $f(x) \equiv g(x) \pmod{p(x)}$

What's the point in assuring that the coefficients of the polynomials belong to $F[x]$ ? Are there some useful properties we can use ?

Bernard
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AleWolf
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    That's not a theorem, that's just a definition. It's the results that come from said definition that might be interesting (though they're more often just generalizations of what you've learned for modular congruences with numbers, so in a way not necessarily too surprising). – PrincessEev May 23 '19 at 07:49

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First of all, $F[x]$ is not a field; $F$ is. And you must have some set to which all coefficients belong. And it's better that they all belong to the same field. Otherwise, what would that definition mean if, say, the coefficients of $f(x)$ belong to $\mathbb C$, whereas the coefficients of $g(x)$ belong to $\mathbb Z/2\mathbb Z$?

José Carlos Santos
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