I'm having a conceptual issue. I know that a splitting field K of p(x) is the smallest field containing both Q and all the roots of p(x)
What about when you aren't given a polynomial? For example:
1) Is $\mathbb{C}$ is a splitting field over $\mathbb{R}$?
2) Is $\mathbb{R}$ is a splitting field over $\mathbb{Q}$?
I am confused how you would begin to answer this. Each field has a lot of different irreducible polynomials.
Thanks!
Best Answer
This is indeed an interesting question. And, yes, there is an intrinsic (or conceptual) characterization of splitting fields of polynomials: they are called normal extensions.
More precisely, given a finite dimensional field extension $F\subset K$ there exists a polynomial $P(X)\in F[X]$ such that $K$ is the splitting field $K=\text{Split}_F(P)$ of $P$ iff the extension $F\subset K$ is normal.
Of course I have to tell you what a normal extension is!
Definition An algebraic extension $F\subset K$ is normal iff every irreducible polynomial $f(X)\in F[X]$ which has a root in $K$ actually is a product of linear factors in $K[X]$.
You then have the generalization of the equivalence above to algebraic but not necessarily finite dimensional extensions $F\subset K$:
$K$ is normal over $F$ iff $K$ is the splitting field $K=\text{Split}_F((P_i)_{i\in I})$ of a (maybe infinite) family $(P_i)_{i\in I}$ of polynomials $P_i(X)\in F(X)$.
For example an algebraic closure $F^{alg}$ of a completely arbitrary field $F$ is clearly normal over $F$ and it is just as clearly a splitting field for the family of all polynomials in $F[X]$ !