<s>[INST] Polynom equality modulo p I found these two equations:
(a) $$X^4 + 1 \equiv (X + 1)^4 \mod \ 2$$ 
(b) $$X^4 + 1 \equiv (X^2 - X - 1)(X^2 + X - 1) \mod \ 3$$
I would like to understand the concept of modulo for Polynoms.
How were they made? And how to verify these equality ?
Thanks in Advanced!
 [/INST] For $(a),$
$y^{m+1}-y=y(y^m-1)$ is divisible by $y(y-1)$ which being the product of two consecutive integers is even
$\implies y^{m+1}\equiv y\pmod2$ for $m\ge1$
$\implies y^4\equiv y\pmod2$ set $y=X, X+1$
For $(b),$
$(x^2-1-x)(x^2-1-x)=(x^2-1)^2-x^2=x^4-3x^2+1\equiv x^4+1\pmod3$
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