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?
Best Answer
Summary of the comments: The splitting field is uniquely determined by the polynomial, if you consider it inside a fixed extension field $E$ of $F$ where the polynomial happens to split. For example, we could use an algebraic closure of $F$ as $E$.
The point of this isomorphism result is to relieve us of the obligation to work inside any previously known field. It allows us to construct the zeros and the extension field any which way we see fit. Secure in the knowledge that the end result will be unique (up to isomorphism).