Let $K=\Bbb{Z}/5\mathbb {Z}$ and $L={\Bbb{Z}/5\mathbb Z[T]\over(T^2+3)}$. Prove that $L$ is a field. Then, let $a=[T+2]\in L$. Prove that a is algebraic over $K$.
I proved that $L$ is a field, since $T^2+3$ has no roots in $\Bbb{Z}/5\mathbb Z$, hence is irreducible. I don't know how to solve the second request, maybe I didn't understand enough the theory.
On
If $K$ is a field and $f(T)\in K[T]$ is an irreducible polynomial of degree $d$, then $K[T]/(f(T))$ is a field containing $K$ and $$ \bigl[K[T]\big/(f(T)):K\bigr]=d $$ Any finite extension of a field is algebraic: indeed, if $u\in F$, a finite extension of $K$ of degree $d$, the powers $\{1,u,u^2,\dots,u^d\}$ form a linearly dependent set over $K$.
We just have to find $f(x)\in K[x]:f(T+2)=0$, where the polynomial itself cancels out using only coefficients from the field $K$ and the equivalence relation $T^3+3=0$.
$$(T+2)^3=T^3+6T^2+12T+8$$ $$(T+2)^2=T^2+4T+4$$ $$(T+2)^1=T+2$$ $$(T+2)^0=1$$
Now we can just use Gaussian elimination on this or something to come up with the coefficients $$f(x)=x^3-6x^2+12x-11$$