$K$ is a splitting field $\iff$ any irreducible polynomial with a root in $K$ splits completely over $K$.
Not sure if there’s a neat direct way. But I learnt that the trick is to add another more conceptual equivalent statement.
Let $K / F$ be a finite extension of fields and let $L$ be an algebraically closed field containing $K$. So we assume $F ⊆ K ⊆ L$. The following are equivalent:
- $K$ is a splitting field for some polynomial $f ∈ F[X]$.
- Every field embedding $σ \colon K → L$ fixing $F$ restricts to $K → K$.
- Every irreducible polynomial $p ∈ F[X]$ with some root $α ∈ K$ splits completely in $K$.
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 $f$.
I have found a direct way proving this:
Suppose $K$ is the splitting field of $g(x)\in F[x]$ and $p(x)\in F[x]$ is irreducible over $F[x]$. Moreover, $p(x)$ has a root in $K$, we want to show that $p(x)$ splits in $K[x]$.
If $p(x)$ is linear, then we are done.
Otherwise, suppose $p(x)=(x-\alpha)(x-\beta)\tilde p(x)$ in the splitting field $\mathcal F$ of $p(x)$, where $\alpha\in K$, $\tilde p(x)\in\mathcal F[x]$ and $\beta\in\mathcal F\setminus K$ is taken to be any root of $p(x)$ that is not in $K$ and we want to get a contradiction.
Note that there is a natural isomorphism between $F(\alpha)$ and $F(\beta)$, since $p(x)$ is irreducible. Therefore, we can extend this isomorphism naturally to an isomorphism between the splitting field of $g(x)$ over $F(\alpha)$ and the splitting field of $g(x)$ over $F(\beta)$. Since $\alpha\in F$ and we conclude that the splitting field of $g(x)$ over $F(\alpha)$ is $K$. If we let $K'$ denote the splitting field of $g(x)$ over $F(\beta)$, then we have $K\cong K'$ and of course $[K':F]=[K:F]$ for their extension degrees.
Note that $K'$ can be viewed as adjoining $\beta$ to $K$ and $\beta\in\mathcal F\setminus K$ implies that $$[K': F]=[K': K][K: F]>[K:F]$$ which is a contradiction.
Thus, no root of $p(x)$ can be taken from $\mathcal F\setminus K$ which implies $p(x)$ splits completely over $K[x]$.
The other direction is simple, please refer to k.stm's answer for "(3)⇒ (1)".