Existence and uniqueness of splitting fields

Let $K$ be a field and let $f(x)\in K[x]$ be a nonconstant polynomial.

Theorem (Splitting fields: existence and uniqueness).

1. (Existence) There exists a field extension $L/K$ such that $f$ factors in $L[x]$ as a product of linear polynomials and $L$ is generated by the roots of $f$, i.e.

   $$L = K(\alpha_1,\dots,\alpha_r)$$

   where $\alpha_1,\dots,\alpha_r\in L$ are all the roots of $f$ in $L$. Such an $L$ is called a splitting field of $f$ over $K$. In particular, $L/K$ is an algebraic extension and is finite.

2. (Uniqueness up to $K$-isomorphism) If $L$ and $L'$ are splitting fields of $f$ over $K$, then there is a $K$-isomorphism $L \cong_K L'$. Equivalently: given a splitting field $L$ of $f$, any $K$-embedding of $L$ into an algebraic closure of $K$ is determined by the images of the roots and must send $L$ onto another splitting field of $f$.

A common construction of $L$ adjoins roots one at a time using simple extensions and then uses the tower law to control degrees.

Examples

1. Over $\mathbb{Q}$, $f(x)=x^2-2$ has roots $\pm\sqrt2$.
   A splitting field is $L=\mathbb{Q}(\sqrt2)$, and $[L:\mathbb{Q}]=2$.

2. Over $\mathbb{Q}$, $f(x)=x^3-2$ has one real root $\sqrt[3]{2}$ and two complex roots $\zeta_3\sqrt[3]{2}$, $\zeta_3^2\sqrt[3]{2}$, where $\zeta_3$ is a primitive cube root of unity.
   A splitting field is $L=\mathbb{Q}(\sqrt[3]{2},\zeta_3)$. (This is also a basic example of a Galois extension once one checks normality and separability.)

3. Over $\mathbb{F}_p$, the polynomial $x^{p^n}-x$ has as its roots exactly the elements of $\mathbb{F}_{p^n}$.
   Its splitting field over $\mathbb{F}_p$ is $\mathbb{F}_{p^n}$, illustrating how finite fields arise as splitting fields.