Let $E = \{a_0+a_1t+a_2t^2|a_i \in \Bbb Z_2 | t^3=1+t \}$, find the roots of $g(x)=x^2+tx+1+t^2$.

Question

Let $E$ be the field defined by $E = \{a_0+a_1t+a_2t^2|a_i \in \Bbb Z_2 | t^3=1+t \}$. I want to find the zeroes of the polynomial $g(x)=x^2+tx+1+t^2$. I am given that they are $t^2, t^2+t$, but I am struggling to verify this is true for $t^2$

Subbing in $g(t)=t^4+t^3+1+t^2=t^4+2(1+t^2)=t^4=t(1+t^2)=t+t^3=-1$.

Here I am stuck, hints appreciated.

Answer 1

$g(t)=t^4+t^3+1+t^2\\=t^4+(1+t)+1+t^2\\=2+t^4+t+t^2=t^4+t(1+t)\\=t^4+t\cdot t^3=2t^4=0$