# How to do congruence-class arithmetic?

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.

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 $$[x^4]=[x^2+x]$$