If we have a polynomial like $f(x)=x^{18}+7x^{13}-2x^5+8$. If we want to consider it mod 5 for example, can we simplify it like by subtracting terms (or adding multiples) in 5's; for example, is $f(x)$ equal to $x^{18}+2x^{13}+3x^5+3$ mod 5? I think I've seen this done before but it feels like this shouldn't be possible. And if it is (or isn't), why is that?
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1Yes its perfectly OK and a frequent practice to simplify the calculations. – Paramanand Singh Mar 23 '21 at 06:25
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1The equality follows via definition of congruence. The difference of old and new polynomial is divisible by $5$. – Paramanand Singh Mar 23 '21 at 06:27
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Thank you, the difference analogy really helped. – Ruochan Liu Mar 23 '21 at 06:28
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You may find helpful this answer – Bill Dubuque Mar 23 '21 at 07:30
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Yes, this is perfectly fine. In you example, because $5\equiv0\pmod{5}$, also $5x^5,5x^{13}\equiv0\pmod{5}$, and hence \begin{eqnarray*} f(x)&=&x^{18}+7x^{13}-2x^5+8\\&\equiv&(x^{18}+7x^{13}-2x^5+8)+(-5x^{13}+5x^5-5)\\ &=&x^{18}+2x^{13}+3x^5+3\pmod{5}. \end{eqnarray*} Similarly, as suggested in the comments, the difference between the two polynomials equals $$-5x^{13}+5x^5-5=5(-x^{13}+x^5-1),$$ which is a multiple of $5$. So the two polynomials are congruent modulo $5$.
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