I tried to prove a proposition from field theory, dealing with the uniqueness of splitting fields and not sure if it's right. Here's my work:
Let $K$ be a field and $f \in K[T]$ a polynominal. Let's say that $L$ and $E$ are splitting fields of $f$. We need to show that $ E \cong L$
We know that every polynominal has a splitting field and that if $f$ is monic ( it could be since $K$ is a field) and $f:=\prod_{\nu=1}^{n}(T-a_{\nu})$, where $a_{1};...;a_{n} \in E$, the splitting field, then $E \cong K[a_{1};...;a_{n}]$ (I'm not going to prove this results here because I've alreday encountered them. Try to prove them if you want)
Induction on $deg(f)$:
$P(1)$: Let $deg(f)=1$. Define $f:=aT+b \in K[T]$. Now, $f=a(T-(-a^{-1}b))$ and hence $K$ is the only splitting field of $f$ and obviously $K \cong K$.
$P(n) \Longrightarrow P(n+1)$: Let $deg(f)=n+1$ and let's assume that for any polynominal of degree $n$, it's splitting fields are isomorphic. Let's define $f:=\sum_{\nu=o}^{n}a_{\nu}T^{\nu}$.
Let $E$ and $L$ be splitting fields of $f$ (again, we assume that $f$ is monic) and define: $f:=\prod_{\nu=1}^{n+1}(T-a_{\nu})=\prod_{\nu=1}^{n+1}(T-c_{\nu})$ with the $a_{\nu}$ terms in $E$ and the $c_{\nu}$ terms in $L$ resp. such that $g:=\prod_{\nu=1}^{n}(T-a_{\nu})=\prod_{\nu=1}^{n}(T-c_{\nu})$.
Now, if we let $E'$ and $L'$ be splitting fields of $g$, since $deg(g)=n$ by the induction hypotesis, $E' \cong L'$. But from the definition of $g$, we deduce that $T-a_{n+1}=T-c_{n+1}$ and therefore $b:=a_{n+1}=c_{n+1}$. From the previous isomorphism, we get that \begin{equation} \begin{split} E'& \cong L' \Leftrightarrow K[a_{1};...;a_{n}] \cong K[c_{1};...;c_{n}] \Longrightarrow E'[b] \cong L'[b] \Leftrightarrow K[a_{1};...;a_{n};b] &\cong K[c_{1};...;c_{n};b] \end{split}\end{equation} but since $b=a_{n+1}=c_{n+1}$, this holds:
$E \cong K[a_{1};...;a_{n};a_{n+1}] \cong K[c_{1};...;c_{n};c_{n+1}] \cong L \Leftrightarrow E \cong L$ which is the final result.
Therefore, since $P(1)$ and the implication $P(n) \Longrightarrow P(n+1)$ hold, $P(n),\, \forall n \in \mathbb{N^{*}}$ holds.
Please answer if I'm right and if I'm not, show me the mistakes.
The problem is that you can have $g\not\in K[T]$. Removing one root you lose control on the coefficients.
The uniqueness of the splitting field is stated as Theorem 3.1 in page 236 of Lang's Algebra. There you'll find an exposition better than I could ever do here.