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I need some help understanding this proof/filling in the details.

I have the following:

We know in general that $[K:F] = [K:K_1][K_1:F]$ but $[K_1:F] = deg(h(x)) \leq n$ (if we consider $K_1$ the splitting field for some $h(x)$.

Apply induction.

$K$ is the splitting field for $f_1(x)$ over $K_1$, and $f_1(x)$ has degree $n − 1$, so its degree is at most $(n − 1)!$.

I'm not sure how this induction works. I assume we're using strong-induction. I would appreciate it if somebody helped me fill in the details of the proof.

Ninosław Brzostowiecki
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    There’s nothing deep about it. Start with $f$ of degree $n$, and find a root $\rho$, giving a field of degree $\le n$ and $f(X)=(X-\rho)f_1(X)$, where the degree of $f_1$ is $n-1$. So the degree of $f_m$ is $n-m$, etc. You should just try it with an $f$ that you know to be irreducible, like $f(X)=X^4+4X+2$. – Lubin Jun 08 '16 at 15:52
  • Related: https://math.stackexchange.com/questions/1017331, https://math.stackexchange.com/questions/62762 – Watson Jun 16 '16 at 12:05

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