How to find degree of separability and degree of inseparability in following question

Question

This question is from an abstract algebra assignment which I am trying.

Let char K = p $\neq 0$ and let $f\in K[x]$ be irreducible of degree n. Let m be the largest nonnegative integer such that f is a polynomial in $x^{p^m}$ but is not a polynomial in $x^{p^{m+1}}$ . Then show that $n = n_op^m $. If u is a root of f, then $[K(u) : K]_s = n_0$ and $[K(u) : K]_i = p^m$.

I have proved that $n =n_0 p^m $ but I am unable to prove that $[K(u):K]_s =n_0$ and $[K(u) : K]_i = p^m$.

I am not sure how should I approach this part and would really appreciate guidence.

Answer 1

One way to approach this structurally is that we want to decompose an arbitrary finite extension into its seperable and inseperable "parts".

So given an arbitrary finite extension $L/K$, one should first show that any two seperable subextensions $L/M_i/K$ are contained in a seperable subextension $L/N/K$. So from this, we see there is a canonical "maximal seperable subextension $L^{sep}$ of $L$ over $K$" associated to any finite extension of fields $L/K$.

Then its not hard to show that $L/L^{sep}$ is purely inseperable, and by the composition law, the degree of $L/K$ is the product of the degrees of $L/L^{sep}$ and $L^{sep}/K$.

So then one just needs to prove that in the case of a primitive extension $K[u]/K$, these degrees have the description you gave. But now we know the constraint that $$[K[u]:K]=[K[u]:K]_i[K[u]:K]_s$$

So we have a few methods of proving the result now, it suffices to prove any of the following:

1. $[K[u]:K]_i=p^m$,
2. $[K[u]:K]_s=n_0$
3. $[K[u]:K]_i\leq p^m$ and $[K[u]:K]_s\leq n_0$
4. $[K[u]:K]_i\geq p^m$ and $[K[u]:K]_s\geq n_0$

I'll leave it up to you to decide which of these methods works for you, they all could be made to work.