<s>[INST] Show that X$^3$+X+1|X$^7$+1 This is a part of the question. Which states Let K= $\mathbb{Z}$/2$\mathbb{Z}$[X]/(d)$\mathbb{Z}$/2$\mathbb{Z}$[X], where
 d=X$^3$+X+1 and let a be the class of X modulo d. This is the only part that i do not know how to solve. I have tried calculating the quotient and the remainder but i do not know if it has any meaningful use. And since it has a remainder i cannot find a q(x) that satisfies f(x)=g(x)*q(x). Any tips?
 [/INST] Here are the steps for polynomial long division $\mod 2$, where the notation
$\dfrac{p(X)}{q(X)} \longrightarrow X^k \tag 0$
indicates the quotient of the leading terms is $X^k$:
$\dfrac{X^7 + 1}{X^3 + X + 1} \longrightarrow X^4; \tag 1$
$X^7 + 1 - X^4(X^3 + X + 1) = X^7 + 1 - X^7 - X^5 - X^4 = X^5 + X^4 + 1; \tag 2$
$\dfrac{X^5 + X^4 + 1}{X^3 + X + 1} \longrightarrow X^2; \tag 3$
$X^5 + X^4 + 1 - X^2(X^3 + X + 1) = X^5 + X^4 + 1 - X^5 - X^3 - X^2 = X^4 + X^3 + X^2 + 1; \tag 4$
$\dfrac{X^4 + X^3 + X^2 + 1}{X^3 + X + 1} \longrightarrow X; \tag 5$
$X^4 + X^3 + X^2 + 1 - X(X^3 + X + 1) = X^4 + X^3 + X^2 + 1 - X^4 - X^2 - X = X^3 + X + 1; \tag 6$
$\dfrac{X^3 + X + 1}{X^3 + X + 1} \longrightarrow 1, \tag 7$
and there is clearly no remainder; we gather the terms of the quotient:
$\dfrac{X^7 + 1}{X^3 + X + 1} = X^4 + X^2 + X + 1, \tag 8$
which is easily checked:
$(X^3 + X + 1)( X^4 + X^2 + X + 1)$
$= X^7 + X^5 + X^4 + X^3 + X^5 + X^3 + X^2 + X + X^4 + X^2 + X + 1$
$X^7 + (X^5 + X^5) + (X^4 + X^4) + (X^3 + X^3) +(X^2 + X^2) + (X + X) + 1$
$=  X^7 + 1; \tag 9$
thus we see that
$X^3 + X + 1 \mid X^7 + 1. \tag{10}$
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