I have this definition for the splitting field of a polynomial: let $f$ be a polynomial with coefficients in the field $F$. A field $E$ containing $F$ is called a splitting field for $f$ if it satisfies:
i) $f$ splits in $E[X]$, i.e $f(X)=a\displaystyle\prod_i (X-\alpha_i)$, with $a,\alpha_i\in E$
ii) $E$ is generated over $F$ by the roots of $f$, i.e. $E=F[\alpha_1,\ldots,\alpha_n]$
Further in my textbook i can find a proof that any two splitting fields for $f$ are $F$-isomorphic.
I can't understand the meaning of this result. By definition, condition ii) seems to state that a splitting field is minimal among all fields containing $F$ where $f$ completely splits, hence i thought that a splitting field was unique. Why then we need a theorem stating that all splitting fields are isomorphic?