[Math] What does it mean to take the splitting field of $f(x)\in F[x]$ over $K$ where $K/F$ is a field extension

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Let $K/F$ be a field extension and let $f(x)\in F[x]$. I know $f(x)$ have a splitting field, i.e. a field $E$ that $f(x)$ splits in ($E/F$ and $f(x)$ doesn't split in any proper subfield of $E$).

I heard the term "the splitting field of $f(x)$ over $K$" – but what does this mean ?

I realize that it is also true that $f(x)\in K[x]$, but I still don't understand the term…it looks like we want a field $L$ s.t $L/K$ is the minimal field extension s.t $f(x)$ splits in $L$ but this like the composition of $E,L$ , but they arn't both subfields of a field I know…

I'm confused, can someone please explain the term ?

Best Answer

Let $K$ be a field and let $f(x)$ be a nonconstant polynomial in $K[x]$. A splitting field of $f$ over $K$ is a field extension $L$ of $K$ such that:

  1. $f(x)$ splits into linear factors in $L$; and
  2. $L = K(\{u\in L\mid f(u) = 0\})$; that is, $L$ is generated over $K$ by the roots of $f(x)$.

One can prove by induction on $\deg(f)$ that splitting fields always exist, and moreover that two splitting fields for $f$ over $K$ are isomorphic over $K$.

One can extend this to arbitrary sets of polynomials: let $K$ be a field, and let $S\subseteq K[x]$ be a set of nonconstant polynomials. A splitting field of $S$ over $K$ is a field extension $L$ of $K$ such that:

  1. For every $f(x)\in S$, $f(x)$ splits in $L$; and
  2. If $U=\{ a\in L \mid \text{there exists }f\in S\text{ such that }f(a)=0\}$, then $L=K(U)$. That is, $L$ is generated over $K$ by the roots of the polynomials in $S$.

It is easy to see that if $S$ is finite, $S=\{f_1,\ldots,f_n\}$, then a splitting field for $S$ over $K$ is the same thing as a splitting field for $g(x)$ over $K$, where $g(x) = f_1(x)\cdots f_n(x)$. Thus, splitting fields are generally only interesting for either single polynomials, or infinite sets of polynomials. The fact that every set of nonconstant polynomials in $K[x]$ has a splitting field over $K$, and moreover that any two such splitting fields are isomorphic over $K$, can be established by applying Zorn's Lemma.

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