Separability in towers - Imported Knowlpedia Content

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

Theorem (tower property for separability).

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

Equivalent characterizations

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 $K$ .

Remarks

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

Examples

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.

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.

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 $L/E$ .