A question about splitting fields. Let $E_{1}$ and $E_{2}$ be two splitting fields of polynomial $p(x)\in \mathbb{F[x]}$ over $\mathbb{F}$. 
My textbook has a long proof for proving that $E_{1}$ and $E_{2}$ are isomorphic. 
But isn't this obvious? Aren't $E_{1}$ and $E_{2}$ equal, as both of them are the smallest field extensions of field $\mathbb{F}$ such that all the roots of $p(x)$ are contained within them?
EDIT: Obviously $E_{1}$ and $E_{2}$ can be constructed differently. But when they're fully constructed, they have to be the very same field! Hence, even if we considered different basis sets for them, $[E_{1}:F]=[E_{2}:F]$. 
 A: Trying to shed some light to this by proffering several splitting fields of the polynomial $x^2+1$ over the rationals.


*

*One splitting field is the field $\mathbb{Q}[i]$, i.e. the usual subfield of complex numbers.

*Another splitting field is the quotient field $\mathbb{Q}[x]/\langle x^2+1\rangle$ of the ring of polynomials with rational coefficients by the maximal ideal generated by the polynomial $x^2+1$.

*Let us consider the skewfield of Hamiltonian quaternions
$$
\mathbb{H}=\{a+bi+cj+dj\mid a,b,c,d\in\mathbb{R}\},
$$
where the usual relations are satisfied: $i^2=j^2=k^2=ijk=-1$. Let us pick any triple $(b,c,d)$ of real numbers such that $b^2+c^2+d^2=1$. Then the quaternion
$$
u=bi+cj+dk
$$
satisfies the equation $u^2=-1$. It is easy to see that the subset
$$
\mathbb{Q}[u]=\{a+fu\in\mathbb{H}\mid a,f\in\mathbb{Q}\}
$$
is then a splitting field of $x^2+1$ over the rationals.

*The set of matrices of the form
$$
\left(\begin{array}{rr}a&b\\-b&a\end{array}\right),
$$
with $a,b$ arbitrary rational numbers is yet another splitting field of $x^2+1$.


No two of these splitting fields are equal in an obvious sense. They are all isomorphic to each other, though.

Another example. You often think of the splitting field of $x^2-2$ over the rationals as $\mathbb{Q}[\sqrt2]$ - a subset of the real numbers. So then $\sqrt2$ may be a certain equivalence class of Cauch sequences. Why is that particular equivalence class of Cauchy sequence equal to the coset of $x$ in the quotient ring $\mathbb{Q}[x]/\langle x^2-2\rangle$?
IOW. This questions is philosophical in the sense that we really are talking about the identity of the roots. Not just up to isomorphism!
