<s>[INST] Let $K=\mathbb{Z}_2[x]/\langle x^4+x^2+x \rangle$ Find a $g\in K$ such that $g^2=x^3+x+1$ Let $K=\mathbb{Z}_2[x]/\langle x^4+x^2+x \rangle$. Find a $g\in K$ such that $g^2=x^3+x+1.$
I tried $x^3+x+1$ itself but unfortunately the degree is only $2$. I don't know how I can multiply something and get a polynomial of degree $3$.
 [/INST] The Quotient ring ${Z_2[x]/(x^4+x^2+x)}$ contains below elements.$${\{1,x,x^2,x^3,x^2+x,x^3+x^2,x^3+x^2+x,x+x^3\}}$$ if we square each of them, then we can see that for ${(x^2+x)^2=x}$, so  ${\sqrt{x}=(x^2+x)=p }$(as defined above by Teresa)

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*now ${(x^3+x+1)^2= x^6+x^2+1= x^3+x^2+x+x^2+1=x^3+x+1}$
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