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When working through this question:

Write out the addition and multiplication tables for the congruence-class ring $F[x]/(p(x))$. $F=\mathbb{Z_2}$; $p(x)=x^{3}+x+1$.

[Question #1 in section 5.2: Congruence-Class Arithmetic of my textbook Abstract Algebra: An Intorduction, 3rd Edition by Thomas W. Hungerford (ISBN: 978-1-1115696-2-4)]

I realized that my answer for the multiplication table did not match the answer in the back of the book. One of the things I am confused on is how $[x^2]*[x^2]=[x^4]=[x^{2}+x]$.

I know that $[x^2]*[x]=[x^3]=[x+1]$ because $[x^3]=(x^{3}+x+1)+(x+1)$ in $\mathbb{Z_2}$. Though when I do this to $[x^2]*[x^2]$ I get $[x^2]*[x^2]=[x^4]=[x^{3}+x+1]$ because $[x^4]=(x^{3}+x+1)+(x^{4}+x^{3}+x+1)$ in $\mathbb{Z_2}$. Which is not correct but I don’t know what I’m missing to make it correct. Any help would be appreciated.

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A better way to see $[x^3]=[x+1]$ is to note that $x^3+x+1=0,$ so $x^3=-x-1=x+1$. Once you have $[x^3]=[x+1]$ you can just multiply both sides by $x$ to get $[x^4]=[x^2+x]$

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