Equivalence of Definitions of Normal Extension

Theorem
Let $L / K$ be an algebraic field extension.

The following statements are logically equivalent:


 * $(1): \quad$ For every irreducible polynomial $f \in K \left[{x}\right]$ with a root in $L$, $f$ splits completely in $L$.


 * $(2): \quad$ For every embedding $\sigma$, of $L$ in the algebraic closure $\overline{K}$ which fixes $K$ pointwise, $\sigma \left({L}\right) = L$.

Proof
$1 \implies 2$

Let $\alpha \in L$ be an arbitrary element.

Let $\sigma: L \mapsto \overline{K}$ be an arbitrary embedding of $L$ fixing $K$.

We wish to show that $\sigma \left({\alpha}\right)\in L$.

Let $m_\alpha$ be the minimal polynomial of $\alpha$ over $K$, which exists because $L / K$ is algebraic.

Since $\sigma$ fixes $K$, $\sigma \left({\alpha}\right)$ must also be a root of $m_\alpha$.

By our assumption, $\alpha\in L$ implies that all roots of $m_\alpha$ are in $L$ and consequently $\sigma \left({\alpha}\right) \in L$.

$2\implies 1$

Again, let $\alpha \in L$ and let $m_\alpha \in K \left[{x}\right]$ be its minimal polynomial over $K$.

We must show that for every root $\beta$ of $m_\alpha$, there exists an embedding $\sigma_\beta$, of $L$ in $\overline{K}$ such that $\sigma_\beta \left({\alpha}\right) = \beta$.

Consider the intermediate field $K \left[{\alpha}\right]\subset L$.

By Abstract Model of Algebraic Extensions, we have an automorphism $\tau_\beta$ for each root $\beta$ of $m_\alpha$ such that $\tau_\beta \left({\alpha}\right) = \beta$ and $\tau_\beta$ fixes $K$

By Extension of Isomorphisms, each $\tau_\beta$ can be extended to an embedding $\sigma_\beta$ of $L$ in $\overline{K}$ such that:
 * $\sigma_\beta \restriction_{K \left[{\alpha}\right]} = \tau_\beta$

By our assumption, $\sigma_\beta \left({L}\right) = L$ for each $\beta$.

Consequently, every root of $m_\alpha$ is in $L$.