Separability in towers

Consider a tower of fields $K\subseteq E\subseteq L$ with $L/K$ algebraic. Separability behaves transitively:

Theorem (tower property for separability).

1. If $L/K$ is a separable extension , then both $E/K$ and $L/E$ are separable.
2. If $E/K$ and $L/E$ are separable, then $L/K$ is separable.

Equivalently at the element level: if $\alpha\in L$ is separable over \(K\) , then it is separable over $E$; conversely, if every element of $E$ is separable over $K$ and every element of $L$ is separable over $E$, then every element of $L$ is separable over $K$.

This interacts cleanly with degree computations via the tower law when the extensions are finite.

Examples

1. Quadratic towers over $\mathbb{Q}$.
   $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt2)\subseteq \mathbb{Q}(\sqrt2,\sqrt3)$ is a finite tower in characteristic $0$, hence both steps and the composite are separable.

2. Purely inseparable tower in characteristic $p$.
   Let $K=\mathbb{F}_p(t)$, $E=K(t^{1/p})$, $L=K(t^{1/p^2})$. Then $E/K$ and $L/E$ are inseparable , and therefore $L/K$ is inseparable as well.

3. Mixed situation.
   If $K$ is perfect (e.g. $K=\mathbb{F}_p$ or $K=\mathbb{Q}$), then any finite $E/K$ is separable (see perfect implies separable ); hence in a finite tower $K\subseteq E\subseteq L$, separability of $L/K$ is equivalent to separability of $L/E$.