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$x^3 \equiv x(mod$ $ 3)$ for all x, whereas obviously $x^3$ and $x$ are not algebraically congruent $(mod$ $ 3)$. What does it mean to be algebraically congruent?. In this case the two polynomials can be viewed as $x^3+0x^2+0x+0 \equiv 0x^3+0x^2+x+0$, so for example the coefficients for $x^3$ are 1 and 0 which are not congruent $(mod$ $ 3)$. Is that right?

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  • $\begingroup$ If you're following a text, what does that text define as two polynomials being algebraically congruent? $\endgroup$ – coffeemath Jul 3 '16 at 23:02
  • $\begingroup$ two polynomials g and f are congruent if the coefficients of each power of x in g and f are congruent (mod m). The text never mentions the word "algebraically" in the definitions, so I infer that's the right definition. $\endgroup$ – TheMathNoob Jul 3 '16 at 23:03
  • $\begingroup$ TheMathNoob In your last comment, what the definition is gives exactly what you spelled out before the final "Is that right?" question. OH, I guess your issue is with the extra qualifier "algebraically"... hard to guess what text intends but I'd guess same as without it. $\endgroup$ – coffeemath Jul 3 '16 at 23:05
  • $\begingroup$ That's what I think it is. $\endgroup$ – TheMathNoob Jul 3 '16 at 23:06

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