Let $K/F$ be a finite extension. I want to show that $K$ is a splitting field over $F$ $\iff$ any irreducible polynomial $p(x)\in F[x]$ that has a root in $K$ splits completely over $K$.
I haven't even been able to show this in a single direction. My efforts have been to show that if $K$ is the splitting field for some set of polynomials $\{f_1,\ldots,f_r\}\subset F[x]$, than any irreducible polynomial $p(x)\in F[x]$ that has a root in $K$ must split completely. Let $p(\alpha)=0$ with $\alpha\in K$. Then we can factor $p(x)$ over $K$: $p(x)=(x-\alpha)p'(x)$.
From here I am actually stuck. It seems showing $p'(x)$ must have a root in $K$ would be useful but isn't really sufficient to show that it splits completely. My progress on that front is as follows. Letting $\beta_1$ and $\beta_2$ be roots of some irreducible (over $F$) factor of one of the $f_i$, we have an automorphism $\phi$ of $K$ obtained by extending the isomorphism $F(\beta_1)\to F(\beta_2)$ to $K$, which in turn extends the identity map on $F$. This fixes $F$ so fixes $p(x)$. Then we have $\phi((x-\alpha)p'(x))=(x-\phi(\alpha))\phi(p'(x))=p(x)$. Thus, if $\phi(\alpha)\neq\alpha$, then we have a new root for $p(x)$ over $K$. But I don't see why one of these automorphisms necessarily doesn't fix $\alpha$, or how we can show $p(x)$ necessarily splits completely this way.
The book (Dummit and Foote) asks this question before introducing any Galois theory, so I would like to solve it without any of those tools. Thank you.
Best Answer
Not sure if there’s a neat direct way. But I learnt that the trick is to add another more conceptual equivalent statement.
Proof. (1) ⇒ (2): Let $f ∈ F[X]$ be a polynomial for which $K$ is a splitting field, say of degree $n$ and monic (without loss of generality). Then $f$ has $n$ roots $α_1, …, α_n ∈ K$ (possibly counted with multiplicites) and $f = (X - α_1)·…·(X - α_n)$ in $K$. Then $K = F(α_1, …, α_n)$, as the latter field is a subextension in which $f$ splits and $K$ is by definition the smallest such extension.
Let $σ \colon K → L$ be any field embedding fixing $F$. Then $σ$ map zeroes of $f$ to zeros of $f^σ = f$, that is: $f(\{α_1,…,α_n\}) = \{α_1,…,α_n\}$. As $σ$ fixes $F$ and $K = F(α_1,…,α_n)$, this implies $σ(K) ⊆ K$.
(2) ⇒ (3): Let $p ∈ F[X]$ be any irreducible polynomial with some root $α ∈ K$. We assume it’s nonzero and monic. Then $p$ splits in $L$ because $L$ is algebraically closed, say $p = (X - α_1)·…·(X - α_n)$ for some $α_1, …, α_n ∈ L$. Have a look at $E = F(α)$. Then we have $n$ maps $σ_1, …, σ_n$ all $E → L$ with $σ_i(α) = α_i$ for $i = 1, …, n$. (For this, we need $p$ to be irreducible!) We can extend those to maps $K → L$, which then have to restricct to maps $K → K$ by assumption. As $α_1, …, α_n$ are in the image of $σ_1, …, σ_n$, we therefore have $α_1, …, α_n ∈ K$ and so $p = (X - α_1)·…(X - α_n)$ in $K[X]$.
(3) ⇒ (1): As a finite extension, there are some $β_1, …, β_n ∈ K$ with $K = F(β_1, …, β_n)$ – for example, take an $F$-basis of $K$. Let $p_1, …, p_n ∈ F[X]$ be the minimal polynomials of $β_1, …, β_n$. Being irreducible, all of them split in $K[X]$, so $f = p_1·…·p_n ∈ F[X]$ does a well and $K$ then is the splitting field of $f$.