Subextensions of Separable Field Extension are Separable
Jump to navigation
Jump to search
Theorem
Let $E / K / F$ be a tower of fields.
Let $E / F$ be separable.
Then $E / K$ and $K / F$ are separable.
Proof
Upper extension
We prove that $E / K$ is separable.
Let $\alpha \in E$.
Let $f$ be its minimal polynomial over $F$.
Let $g$ be its minimal polynomial over $K$.
Then by hypothesis, $f$ is separable.
On the other hand:
- $f \in K \sqbrk x$
and:
- $\map f \alpha = 0$
Hence by definition $g$ divides $f$.
This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: The following proves Divisor of Separable Polynomial is Separable You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page. |
Consider the roots of $f$ and $g$ in the algebraic closure of $K$.
Since $f$ is separable, it has no repeated roots.
Since $g$ divides $f$, the roots of $g$ are included in the roots of $f$,
we see that $g$ has no repeated roots in the algebraic closure of $K$,
so $g$ is separable.
$\Box$
Lower extension
It follows immediately by definition that $K / F$ is a separable extension.
$\blacksquare$
Also see
- Transitivity of Separable Field Extensions, the converse
Source
- 2002: Serge Lang: Algebra (Revised 3rd ed.): Chapter V, $\S4$ Separable Extensions, the paragraph before Theorem $4.3$