Jacobson radical of $\mathbb{F}_{2}(t)[x]/(x^4-t^2)$ Let $\mathbb{F}_{2}$ be the field of two elements. Let $R=\mathbb{F}_{2}(t)[x]/(x^4-t^2)$.
Why is $R/J(R)$ equal to $\mathbb{F}_{2}(t)[x]/(t-x^2)$? here $J(R)$ denotes the Jacobson radical of $R$.

The Jacobson radical of $R$ is, by definition, the intersection of maximal ideals of $R$. In our case $R$ is local with the maximal ideal $(x^2+t)/(x^2+t)^2$. If we want to see this as an $R$-module, then we can conclude that it is isomorphic to $R/(x^2+t)$, but I can't see how is this isomorphic to $\mathbb F_2(t)[x]/(x^2+t)$ (which is actually isomorphic to $R/J(R)$). 
