Is $F_1$ a subfield of $K_f$?

Question

Let $F$ be a subfield of $\mathbb C$ and let $f \in F[x]$. Let $K_f=$ the splitting field for $f$ over $F$. Suppose $F_1/F$ is a Galois extension of prime degree. Is $F_1$ a subfield of $K_f$?

FYI, I am attempting to prove a lemma to help me solve Prove that if the Galois group of a polynomial $f$ is a nonabelian simple group, then the roots are not solvable.)

Answer 1

With the givens, you shouldn't expect $f$ and $F_1$ to be related. For example, take:

• $F = \mathbb{Q}$
• $f = x$
• $F_1 = \mathbb{Q}(i)$

Then $K_f = \mathbb{Q}$, which not an overfield of $F_1$ (whether you ask the question abstractly or as subfields of $\mathbb{C}$).

If you want an example where $K_f$ is a nontriival extension, take $f = x^2 - 2$.