Please check these statements whether those are true
Let $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ be extension fields of $\mathbb{Q}$, the rational numbers. Since they are not isomorphic as field hence 'If $K$ and $E$ have isomorphic Galois group over $F$, then $K$ and $E$ are isomorphic as field' is false.
Let $K$, $E$ be splitting fields of polynomial $x^2+3$ , $x^3-1$ over $\mathbb{Q}$ , respectively. Actually, their splitting fields are same. It's $\mathbb{Q}(i\sqrt{3})$. Therefore, 'If $K$ and $E$ have same splitting field even if they are splitting fields of different polynomials over same field $F$, then $K$ and $E$ have isomorphic Galois group over $F$' is false.
The reason number 2. is that $E$ has Galois group $\mathbb{Z}_3$ (in fact $E$'s splitting field is $\mathbb{Q}(\omega)$ where $\omega$ is a root of $x^2+x+1$ hence automorphism of $E$ depends on value of $\omega$ and it can be $\omega$ or $\omega^2$ or $1$) but $K$ has $\mathbb{Z}_2$.