Irreducible polynomial that has a multiple zero example

Question

I'm probably missing a point in this example, but what does the zero of f(x) look like in $F[x]/\langle x^2-t\rangle$?

Consider $x^2-t\in F[x]/\langle x^2-t\rangle$. Then the roots of $f(x)$ are $\pm\sqrt{t}$, each of multiplicity $1$, and each not belonging to $\mathbb{Z}_2(t)$. What point(s) am I missing here?

Answer 1

I guess the point you are missing is that $\sqrt{t}=-\sqrt{t}$ because $F[x]$ and its field of fractions has characteristic $2$. There is actually only one root.