I have the following problems:
(1) Let $g=X^2+\overline{4}$ and $h=X^2+\overline{2}$ be polynomials in $(\mathbb{Z}/\mathbb{Z}7)[X]$. $L$ and $K$ are the splitting fields of $g$ and $h$ over $\mathbb{Z}/\mathbb{Z}7$.
Find a field isomorphism $$ \phi \colon \ L \longrightarrow K$$ an proove it is one.
(2) Furthermore I have to show, that the splitting fields $L$ of $X^2-2$ and $K$ of $X^2-3$ (over $\mathbb{Q}$) are not isomorph.
Remember the splitting fields of a quadratic polynomial $f(t)$ over a field $F$ all look like $\{ax+b\mid a,b\in F\}$ for some $x$ satisfying $f(x)=0$. The fact that $f(x)=0$ shows you how to multiply two elements of the form $ax+b$ to obtain another element of the same form.
Hints:
(1) In $L$, we have an element $\alpha$ whose square is $-4\equiv 3\pmod 7$, and in $K$ we have an element $\beta$ whose square is $-2\equiv 5\pmod 7$. What is the square of $2\alpha$ in $L$?
(2) If the splitting fields were isomorphic, then the splitting field of $X^2-3$ would have an element whose square is $2$.