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I was told by my tutor to interpret the notation $$ f(x) \equiv g(x) \hspace{2mm} (\text{mod} \hspace{2mm} h(x), \hspace{1mm} n) $$ as meaning that "$f(x)$ is equivalent to $g(x)$ in the polynomial quotient ring $\mathbb{Z}_n [x] / h(x)$".

Assuming that this is a correct definition of this notation, I am wondering how exactly to interpret this. Am I correct in thinking that this means that $$ f(x) \in \left\{ \hspace{1mm} g(x) \cdot h(x)^k \hspace{2mm} | \hspace{2mm} k \in \mathbb{N} \hspace{1mm} \right\} $$

For example, would $$ f(x) \equiv (x^2 + 1) \hspace{2mm} (\text{mod} \hspace{2mm} (x^4 + 1), \hspace{1mm} 2) $$ mean that $$ f(x) \in \left\{ (x^2 + 1), \hspace{1mm} (x^2 + 1)(x^4 + 1), \hspace{1mm} (x^2 + 1)(x^4 + 1)^2, \hspace{1mm} (x^2 + 1)(x^4 + 1)^3, \dots \hspace{1mm} \right\} $$

M Smith
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1 Answers1

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Your tutor's interpretation is completely correct, but perhaps a little needlessly convoluted if you're very new to ring theory. All it means is: there exist some polynomials $q(x)$ and $r(x)$ such that $$f(x) = g(x) + (h(x)q(x) + nr(x)).$$ In other words, exactly as when working $``\mod m"$ when $m$ is an integer, you are allowed to add and subtract any multiples of $h(x)$ and $n$ as you like.

Billy
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