Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible polynomial of $u$ in $K$ is separable.

A long way to see this is to use the fact that $F$ is algebraic and Galois over $K$, but I want give a direct proof without talking about Galois.


Look at the Old Answer by Pierre-Yves Gallaird to Splitting field of a separable polynomial is separable. He argues that the following three statements are equivalent for a field extension $L/K$ of degree $d < \infty$:

  1. $L/K$ is generated by separable elements;
  2. Every element of $L$ is separable over $K$;
  3. There are exactly $d$ embeddings of $L$ in $\overline L = \overline K$ over $K$.

For your case, you'd additionally have to argue that an element $u \in F$ is already contained in the splitting field $L \subseteq F$ of a finite subset of $S$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.