<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] We use the identity $(u+v)^2=u^2+v^2.$ From $$x^4+x^2+x=0$$ we see that
$$x=x^4+x^2=(x^2)^2+x^2=(x^2+x)^2$$
And we can construct the root of an arbitrary polynomial
$$ax^3+bx^2+cx+d=x(ax^2+c)+bx^2+d=(x^2+x)^2(ax+c)^2+(bx+d)^2=((x^2+x)(ax+c)+(bx+d))^2=(ax^3+(a+c)x^2+(b+c)x+d)^2$$
So we see that every polynomial has exactly one square root.
</s>